equality - The state of being equal. A relationship between two quantities or two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object

-3x<= -9 or 5+x<6

-3x<= -9 or 5+x<6
Take each piece:
-3x<= -9
Divide each side by -3:
x>=3
Now take 5 + x < 6
5 + x < 6
Subtract 5 from each side:
x < 1
Joining together the two inequalities, we have:
x<1 or x>=3
Use our [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=x%3C1orx%3E%3D3&pl=Show+Interval+Notation']interval notion calculator[/URL] to find the interval notation of this compound inequality

12 plus the product of 4 and a number is greater than 72

A number means an arbitrary variable, let's call it x.
The product of 4 and a number is 4x.
12 plus that product is 4x + 12
Is greater than means an inequality, so we set the entire expression greater than 72
4x + 12 > 72

13 more than x is greater than 14

13 more than x means we add:
x + 13
This expression is greater than 14, so we write an inequality:
x + 13 > 14

2 is greater than or equal to w and -7 is less than or equal to w

2 is greater than or equal to w and -7 is less than or equal to w
Written as an inequality, we have:
-7 <= w <= 2

2/3 of a number 17 is at least 29

2/3 of a number 17 is at least 29
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
2/3 of a number means we multiply x by 2/3:
2x/3
The phrase [I]is at least[/I] also means greater than or equal to, so we set up the inequality:
[B]2x/3 >= 29[/B]

3 times a number increased by 1 is between -8 and 13

3 times a number increased by 1 is between -8 and 13.
Let's take this algebraic expression in [U]4 parts[/U]:
Part 1 - The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Part 2 - 3 times this number means we multiply x by 3:
3x
Part 3 - Increased by 1 means we add 1 to 3x:
3x + 1
The phrase [I]between[/I] means we have an inequality:
[B]-8 <= 3x + 1 <=13[/B]

35 added to n is greater than or equal to the sum of k and 21

35 added to n is greater than or equal to the sum of k and 21
Take this algebraic expression in 3 parts:
[LIST=1]
[*]35 added to n means we have a sum: n + 35
[*]The sum of k and 21 means we add 21 to k: k +21
[*]The phrase [I]greater than or equal to[/I] means an inequality using this sign (>=), so we write this as follows:
[/LIST]
[B]n + 35 >= k + 21[/B]

50 is more than the product of 4 and w

50 is more than the product of 4 and w
Take this algebraic expression in pieces:
The product of 4 and w mean we multiply the variable w by 4:
4w
The phrase [I]is more than[/I] means an inequality using the (>) sign, where 50 is greater than 4w:
[B]50 > 4w[/B]

6 diminished by twice x is at most 8

6 diminished by twice x is at most 8
Twice x means we multiply x by 2:
2x
6 diminished by twice x means we subtract 2x from 6:
6 - 2x
The phrase [I]is at most[/I] is an inequality using the sign <=, so we have:
[B]6 - 2x <= 8[/B]

6 times a number, x, is at least 22.

6 times a number, x, is at least 22.
6 times a number x:
6x
The phrase [I]is at least[/I] means greater than or equal to. So we have an inequality:
[B]6x >= 22[/B] <-- This is our algebraic expression
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=6x%3E%3D22&pl=Show+Interval+Notation']To solve this for x, paste this into the search engine[/URL] and we get:
[B]x >= 3.666667[/B]

7 times a positive number n is decreased by 3, it is less than 25

7 times a positive number n is decreased by 3, it is less than 25
7 times a positive number n:
7n
Decreased by 3:
7n - 3
The phrase [I]it is less than [/I]means an inequality. So we relate 7n - 3 less than 25 using the < sign to get our algebraic expression of:
[B]7n - 3 < 25[/B]

8 more than twice a number is less than 6 more than the number

8 more than twice a number is less than 6 more than the number.
This is an algebraic expression, let's take it in pieces...
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
8 more than twice a number:
Twice a number means multiply x by 2: 2x
Then add 8: 2x + 8
6 more than the number, means we add 6 to x
x + 6
The phrase [I]is less than[/I] means an inequality, where we set 2x + 8 less than x + 6
[B]2x + 8 < x + 6[/B]

9 is greater than x and x is greater than 3

9 is greater than x and x is greater than 3
This is a compound inequality. We write this as:
[B]3 < x < 9[/B]

9 subtracted from the product of 3 and a number is greater than or equal to 16

9 subtracted from the product of 3 and a number is greater than or equal to 16
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x
[*]The product of 3 and a number means we multiply 3 times x: 3x
[*]9 subtracted from the product: 3x - 9
[*]The phrase is greater than or equal to means an inequality. So we set up an inequality with >= for the greater than or equal to sign in relation to 3x - 9 and 16
[/LIST]
Our algebraic expression (inequality) becomes:
[B]3x - 19 >= 16[/B]

A bakery has a fixed cost of $119.75 per a day plus $2.25 for each pastry. The bakery would like to

A bakery has a fixed cost of $119.75 per a day plus $2.25 for each pastry. The bakery would like to keep its daily costs at or below $500 per day. Which inequality shows the maximum number of pastries, p, that can be baked each day.
Set up the cost function C(p), where p is the number of pastries:
C(p) = Variable Cost + Fixed Cost
C(p) = 2.25p + 119.75
The problem asks for C(p) at or below $500 per day. The phrase [I]at or below[/I] means less than or equal to (<=).
[B]2.25p + 119.75 <= 500[/B]

A bank charges a service fee of $7.50 per month for a checking account. A bank account has $85.00. I

A bank charges a service fee of $7.50 per month for a checking account. A bank account has $85.00. If no money is deposited or withdrawn except the service charge, how many months until the account balance is negative?
Let m be the number of months. Our balance is denoted by B(m):
B(m) = 85 - 7.5m
The question asks when B(m) is less than 0. So we set up an inequality:
85 - 7.5m < 0
To solve this inequality for m, [URL='https://www.mathcelebrity.com/1unk.php?num=85-7.5m%3C0&pl=Solve']we type it in our search engine[/URL] and we get:
m > 11.3333
We round up to the next whole integer and get [B]m = 12[/B]

a baseball park charges $4.50 per admission ticket. the park has already sold 42 tickets. how many m

a baseball park charges $4.50 per admission ticket. the park has already sold 42 tickets. how many more tickets would they need to sell to earn at least $441?
Let the number of tickets above 42 be t.
A few things to note on this question:
[LIST]
[*]The phrase [I]at least[/I] means greater than or equal to, so we have an inequality.
[*]Earnings = Price * Quantity
[/LIST]
We're given:
Earnings = 4.50 * 42 + 4.5t >= 441
Earnings = 189 + 4.5t >= 441
We want to solve this inequality for t:
Solve for [I]t[/I] in the inequality 189 + 4.5t ? 441
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 189 and 441. To do that, we subtract 189 from both sides
4.5t + 189 - 189 ? 441 - 189
[SIZE=5][B]Step 2: Cancel 189 on the left side:[/B][/SIZE]
4.5t ? 252
[SIZE=5][B]Step 3: Divide each side of the inequality by 4.5[/B][/SIZE]
4.5t/4.5 ? 252.4.5
[B]t ? 56[/B]

A bus ride cost 1.50. A 30 day pass cost $24. Write an inequallity to show that the 30 day pass is t

A bus ride cost 1.50. A 30 day pass cost $24. Write an inequallity to show that the 30 day pass is the better deal
Let the number of days be d. We have the inequality below where we show when the day to day cost is greater than the monthly pass:
1.5d > 24
To solve this inequality for d, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=1.5d%3E24&pl=Show+Interval+Notation']type it in our search engine[/URL] and we get:
[B]d > 16[/B]

A car salesman earns $800 per month plus a 10% commission on the value of sales he makes for the mon

A car salesman earns $800 per month plus a 10% commission on the value of sales he makes for the month. If he is aiming to earn a minimum of $3200 a month, what is the possible value of sales that will enable this?
to start, we have:
[LIST]
[*]Let the salesman's monthly sales be s.
[*]With a 10% discount as a decimal of 0.1
[*]The phrase [I]a minimum[/I] also means [I]at least[/I] or [I]greater than or equal to[/I]. This tells us we want an inequality
[*]We want 10% times s + 800 per month is greater than or equal to 3200
[/LIST]
We want the inequality:
0.1s + 800 >= 3200
To solve for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.1s%2B800%3E%3D3200&pl=Solve']type this inequality into our search engine[/URL] and we get:
[B]s >= 24000[/B]

A car who’s original value was $25600 decreases in value by $90 per month. How Long will it take bef

A car who’s original value was $25600 decreases in value by $90 per month. How Long will it take before the cars value falls below $15000
Let m be the number of months.We have our Book Value B(m) given by:
B(m) = 25600 - 90m
We want to know when the Book value is less than 15,000. So we have an inequality:
25600 - 90m < 15000
Typing [URL='https://www.mathcelebrity.com/1unk.php?num=25600-90m%3C15000&pl=Solve']this inequality into our search engine and solving for m[/URL], we get:
[B]m > 117.78 or m 118 months[/B]

a carnival charges $6 admission and $2.50 per ride. You have $50 to spend at the carnival. Which of

a carnival charges $6 admission and $2.50 per ride. You have $50 to spend at the carnival. Which of the following inequalities represents the situation if r is the number of rides?
We set up our inequality using less than or equal to, since our cash is capped at $50. We use S for our :
Cost per ride * r + Admission <= 50
Plugging in our numbers, we get:
2.50r + 6 <= 50
[B][/B]
Now, if the problem asks you to put this in terms of r, then [URL='https://www.mathcelebrity.com/1unk.php?num=2.50r%2B6%3C%3D50&pl=Solve']we plug this inequality into our search engine[/URL] and we get:
r <= 17.6
Since we cannot do fractional rides, we round down to 17:
[B]r <= 17[/B]

A carnival charges a $15 admission price. Each game at the carnival costs $4. How many games would a

A carnival charges a $15 admission price. Each game at the carnival costs $4. How many games would a person have to play to spend at least $40?
Let g be the number of games. The Spend function S(g) is:
S(g) = Cost per game * number of games + admission price
S(g) = 4g + 15
The problem asks for g when S(g) is at least 40. At least is an inequality using the >= sign:
4g + 15 >= 40
To solve this inequality for g, we type it in our search engine and we get:
g >= 6.25
Since you can't play a partial game, we round up and get:
[B]g >= 7[/B]

A certain species of fish costs $3.19 each. You can spend at most $35. How many of this type of f

A certain species of fish costs $3.19 each. You can spend at most $35. How many of this type of fish can you buy for your aquarium?
Let the number of fish be f. We have the following inequality where "at most" means less than or equal to:
3.19f <= 35
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=3.19f%3C%3D35&pl=Show+Interval+Notation']Typing this inequality into our search engine[/URL], we get:
f <= 10.917
Since we need a whole number of fish, we can buy a maximum of [B]10 fish[/B].

a company has revenue given by R(x)=500x dollars and total cost given by C(x)=48,000 100x dollars, w

a company has revenue given by R(x)=500x dollars and total cost given by C(x)=48,000 + 100x dollars, where x is the number of units produced and sold. How many units will give a profit
Profit P(x) is given by:
R(x) - C(x)
So we have:
P(x) = 500x - (100x + 48,000)
P(x) = 500x - 100x - 48,000
P(x) = 400x - 48,000
A profit is found when P(x) > 0, so we have:
400x - 48000 > 0
To solve this inequality, [URL='https://www.mathcelebrity.com/1unk.php?num=400x-48000%3E0&pl=Solve']we type it into our search engine [/URL]and we get:
[B]x > 120[/B]

A cup of coffee costs $1.75. A monthly unlimited coffee card costs $25.00. Which inequality represe

A cup of coffee costs $1.75. A monthly unlimited coffee card costs $25.00. Which inequality represents the number x of cups of coffee you must purchase for the monthly card to be a better deal?
Let c be the number of cups. We want to know how many cups (x) where:
1.75x > 25
Using our [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=1.75x%3E25&pl=Show+Interval+Notation']inequality solver[/URL], we see:
[B]x > 14.28[/B]

A local bank charges 19 per month plus 3 cents per check. The credit union charges7 per month plus

A local bank charges 19 per month plus 3 cents per check. The credit union charges7 per month plus 7 cents per check. How many checks should be written each month to make the credit union a better deal?
Set up the cost function B(c) for the local bank where c is the number of checks:
B(c) = 0.03c + 19
Set up the cost function B(c) for the credit union where c is the number of checks:
B(c) = 0.07c + 7
We want to find out when:
0.07c + 7 < 0.03c + 19
[URL='https://www.mathcelebrity.com/1unk.php?num=0.07c%2B7%3C0.03c%2B19&pl=Solve']Typing this inequality into our search engine[/URL], we get:
c < 300

A local sports centre charges $8 per visit. For an annual membership fee of$45, the cost per visit i

A local sports centre charges $8 per visit. For an annual membership fee of$45, the cost per visit is only $5.50. What is the least number of visits needed in a year in order for the membership to be a better deal?
Set up the cost for the visitors plan C(v) where v is the number of visits:
C(v) = 8v
Set up the cost for the membership plan C(v) where v is the number of visits:
C(v) = 5v + 45
The problem asks for v where:
5v + 45 < 8v
[URL='https://www.mathcelebrity.com/1unk.php?num=5v%2B45%3C8v&pl=Solve']Type this inequality into our search engine[/URL] and get:
v > 15
This means, the least number of visits is 1 more which is [B]16[/B]

A Middleweight UFC fighter weighs between 170 lbs and 185 lbs.

A Middleweight UFC fighter weighs between 170 lbs and 185 lbs.
Let w be the UFC fighter's weight:
We have a compound inequality.
Right side includes 185 lbs. because between means includes 185lbs.
Left side includes 170 lbs. because between means includes 17lb0s
[B]170 <= w <= 185[/B]

a more than b is greater than 6

a more than b is greater than 6
a more than b:
b + a
Is greater than 6 means an inequality using the > sign:
[B]b + a > 6[/B]

A number n is no less than 2 and no more than 49.

A number n is no less than 2 and no more than 49.
This is a compound inequality. Let's break it into parts.
Step 1: No more than 49 means 49 or less. Or, less than or equal to 49
<= 49
Step 2: no less than 2 means 2 or greater. Or, greater than or equal to 2
>=2
Writing this in terms of the number n, we have:
[B]2 <= n <= 49[/B]

A number t is no less than 30 and fewer than 40.

A number t is no less than 30 and fewer than 40.
This is a compound inequality. Take it in 3 parts:
Step 1: fewer than 40 means less than (does not include 40)
t < 40
Step 2: no less than 30 means greater than or equal to
t >= 30
Step 3: Combine these 2 statements into one compound inequality:
[B]30 <= t < 40[/B]

A phone company offers two monthly charge plans. In Plan A, there is no monthly fee, but the custome

A phone company offers two monthly charge plans. In Plan A, there is no monthly fee, but the customer pays 8 cents per minute of use. In Plan B, the customer pays a monthly fee of $1.50 and then an additional 7 cents per minute of use.
For what amounts of monthly phone use will Plan A cost more than Plan B?
Set up the cost equations for each plan. The cost equation for the phone plans is as follows:
Cost = Cost Per Minute * Minutes + Monthly Fee
Calculate the cost of Plan A:
Cost for A = 0.08m + 0. <-- Since there's no monthly fee
Calculate the cost of Plan B:
Cost for B = 0.07m + 1.50
The problem asks for what amounts of monthly phone use will Plan A be more than Plan B. So we set up an inequality:
0.08m > 0.07m + 1.50
[URL='https://www.mathcelebrity.com/1unk.php?num=0.08m%3E0.07m%2B1.50&pl=Solve']Typing this inequality into our search engine[/URL], we get:
[B]m > 150
This means Plan A costs more when you use more than 150 minutes per month.[/B]

A quantity x is at least 10 and at most 20

A quantity x is at least 10 and at most 20
The phrase [I]at most[/I] means less than or equal to
The phrase [I]at least[/I] means greater than or equal to.
So we have the following inequality
[B]10 <= x <= 20[/B]

A quarter of a number is greater than or equal to 38

A quarter of a number is greater than or equal to 38.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
A quarter of a number means 1/4, so we have:
x/4
The phrase [I]is greater than or equal to[/I] means an inequality, so we use the >= sign in relation to 38:
[B]x/4 >= 38 <-- This is our algebraic expression
[/B]
If you want to solve this inequality, [URL='https://www.mathcelebrity.com/prop.php?num1=x&num2=38&propsign=%3E%3D&den1=4&den2=1&pl=Calculate+missing+proportion+value']we type it in the search engine[/URL] to get:
x >= [B]152[/B]

A river is rising at a rate of 3 inches per hour. If the river rises more than 2 feet, it will excee

A river is rising at a rate of 3 inches per hour. If the river rises more than 2 feet, it will exceed flood stage. How long can the river rise at this rate without exceeding flood stage?
Let the number of inches be i. Remember 12 inches to a foot, so we have 2 feet = 12*2 = 24 inches.
[LIST]
[*]Inequality: 3i <= 24. (since more than means the river can go [U]up to[/U] 2 feet or 24 inches
[/LIST]
To solve the inequality for I, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=3i%3C%3D24&pl=Show+Interval+Notation']type it in our search engine[/URL] and we get:
[B]I <= 8
This means after 8 hours, the river will flood[/B]

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to s

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to spend less than $12 on a ride. Which inequality can be used to find the distance Samantha can travel?
Set up the travel cost equation where m is the number of miles:
C(m) = 0.8m + 1.50
If Samantha wants to spend less than 12 per ride, we have an inequality where C(m) < 12:
[B]0.8m + 1.50 < 12[/B]

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to s

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to spend less than $12 on a ride. Which inequality can be used to find the distance Samantha can travel?
[LIST]
[*]Each ride will cost 1.50 + 0.8x where x is the number of miles per trip.
[*]This expression must be less than 12.
[/LIST]
[U]Setup the inequality:[/U]
1.5 + 0.8x < 12
[U]Subtracting 1.5 from each side of the inequality[/U]
0.8x < 10.5
[U]Simplifying even more by dividing each side of the inequality by 0.8, we have:[/U]
[B]x < 13.125[/B]

a total of one and x is less than 5

a total of one and x is less than 5
A total of one and x means we add x to 1:
1 + x
We set up an inequality, where 1 + x is less than (<) 5
[B]1 + x < 5[/B]

absolute value of x is less than or equal to 4

absolute value of x is less than or equal to 4
Absolute value of x:
|x|
Set up an inequality where this is less than or equal to 4:
[B]|x| <= 4 [/B] <-- This is our algebraic expression
To solve this, we have the following compound inequality:
-4 < x < 4

Addition Equality Property

Demonstrates the Addition Equality Property
Numerical Properties

Addition Property Of Inequality

Demonstrates the Addition Property Of Inequality.
Numerical Properties

Ali runs each lap in 6 minutes. He will run at least 11 laps today. What are the possible numbers of

Ali runs each lap in 6 minutes. He will run at least 11 laps today. What are the possible numbers of minutes he will run today?
Let m be the number of minutes. The phrase [I]at least[/I] means an inequality, also known as greater than or equal to. So we have:
m >= 11*6
[B]m >= 66
You can read this as Ali will run 66 or more minutes today. Or at least 66 minutes. Or greater than or equal to 66 minutes[/B]

Amanda runs each lap in 4 minutes. She will run less than 44 minutes today. What are the possible nu

Amanda runs each lap in 4 minutes. She will run less than 44 minutes today. What are the possible number of laps she will run today?
Notes for this problem:
[LIST]
[*]Let laps be l.
[*]Lap time = Time per lap * number of laps (l)
[*]Less than means we have an inequality using the < sign
[/LIST]
We have the inequality:
4l < 44
To solve this inequality for l, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=4l%3C44&pl=Show+Interval+Notation']type it in our math engine[/URL] and we get:
[B]l < 11[/B]

An Amazon delivery package receives a bonus if he delivers a package to a costumer in 20 minutes plu

An Amazon delivery package receives a bonus if he delivers a package to a costumer in 20 minutes plus or minus 5 minutes. Which inequality or equation represents the drivers allotted time (x) to receive a bonus
20 plus 5 minutes = 25 minutes
20 minus 5 minutes = 15 minutes
So we have the inequality:
[B]15 <= x <= 25[/B]

An avocado is not ripe until 4 days after picking and will go bad after 7 days after picking. Repres

An avocado is not ripe until 4 days after picking and will go bad after 7 days after picking. Represent the days the avocado is ripe
Our sweet spot for ripeness is 4 days or more but not more than 7 days. Using d as our days, we have the following inequality:
[B]4 <= d <= 7[/B]

An elevator can hold less than 2700 pounds of extra weight. If an average person weighs 150 pounds,

An elevator can hold less than 2700 pounds of extra weight. If an average person weighs 150 pounds, what is the maximum number of people (p) that can be on the elevator at one time?
Total weight = average weight per person * Number of people
Total weight = 150p
We know from the problem that:
150p < 2700
We want to solve this inequality for p. Divide each side of the inequality by 150:
150p/150 < 2700/150
Cancel the 150's on the left side and we get:
p < [B]18[/B]

An elevator has a maximum weight of 3000 pounds. How many 150-pound people can safely ride the eleva

An elevator has a maximum weight of 3000 pounds. How many 150-pound people can safely ride the elevator? (Use "p" to represent the number of people)
Maximum means less than or equal to. We have the inequality:
150p <= 3000
To solve this inequality for p, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=150p%3C%3D3000&pl=Show+Interval+Notation']type it in our search engine[/URL] and we get:
[B]p <= 20[/B]

Ana's height is strictly between 63 and 66 inches. Write a symbolic inequality to represent this sce

Ana's height is strictly between 63 and 66 inches. Write a symbolic inequality to represent this scenario. let h be height
[B]63 < h < 66
[/B]
You can also type [I][URL='https://www.mathcelebrity.com/algexpress.php?num=between63and66&pl=Write+Expression']between 63 and 66[/URL][/I] in our search engine.

Angelica’s financial aid stipulates that her tuition cannot exceed $1000. If her local community col

Angelica’s financial aid stipulates that her tuition cannot exceed $1000. If her local community college charges a $35 registration fee plus $375 per course, what is the greatest number of courses for which Angelica can register?
We set up the Tuition function T(c), where c is the number of courses:
T(c) = Cost per course * c + Registration Fee
T(c) = 35c + 375
The problem asks for the number of courses (c) where her tuition [I]cannot exceed[/I] $1000. The phrase [I]cannot exceed[/I] means less than or equal to, or no more than. So we setup the inequality for T(c) <= 1000 below:
35c + 375 <= 1000
To solve this inequality for c, we [URL='https://www.mathcelebrity.com/1unk.php?num=35c%2B375%3C%3D1000&pl=Solve']type it in our search engine and we get[/URL]:
c <= 17.85
Since we cannot have fractional courses, we round down and get:
c[B] <= 17[/B]

As a salesperson you will earn $600 per month plus a commission of 20% of sales. Find the minimum am

As a salesperson you will earn $600 per month plus a commission of 20% of sales. Find the minimum amount of sales you need to make in order to receive a total income of at least $1500 per month.
Let the amount of sales be s. The phrase [I]at least[/I] means greater than or equal to. Since 20% is 0.2, We want to know when:
0.20s + 600 >= 1500
We [URL='https://www.mathcelebrity.com/1unk.php?num=0.20s%2B600%3E%3D1500&pl=Solve']type this inequality into our search engine to solve for s[/URL] and we get:
s >= [B]4500[/B]

As a salesperson, you are paid $50 per week plus $2 per sale. This week you want your pay to be at l

As a salesperson, you are paid $50 per week plus $2 per sale. This week you want your pay to be at least $100. What is the minimum number of sales you must make to earn at least $100?
Set up the inequality where s is the amount of sales you make:
50 + 2s >= 100
We use >= because the phrase [I]at least[/I] 100 means 100 or more
Subtract 50 from each side:
2s >= 50
Divide each side by 2
[B]s >= 25[/B]

ason decided that he will sell his stocks if their values per share (x) goes below $5 or above $15.

Jason decided that he will sell his stocks if their values per share (x) goes below $5 or above $15. Write a compound inequality represents the values at which Jason will sell his stocks?
Below $5 is also known as less than $5:
x < 5
Above $15 is also known as greater than $15
x > 15
We write the compound inequality:
[B]x < 5 U x > 15[/B]

Beverly has $50 to spend at an amusement park. She plans to spend $10 for food, and $15 for admissio

Beverly has $50 to spend at an amusement park. She plans to spend $10 for food, and $15 for admission to the park. Each ride costs $1.50 to ride. Write an inequality to represent the possible number of rides she can ride?
First, we subtract the food and admission cost from Beverly's starting balance of $50:
Cost available for rides = Starting Balance - Food - Admission
Cost available for rides = 50 - 10 - 15
Cost available for rides = 25
Now we set up an inequality for the number of rides (r) that Beverly can ride with the remaining balance:
1.50r <= 25
To solve for r, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=1.50r%3C%3D25&pl=Show+Interval+Notation']type this inequality into our search engine[/URL] and we get:
[B]r <=[/B] [B]16.67[/B]

Bob weighs between 125 and 135

Bob weighs between 125 and 135
Let w be Bob's weight. Between means includes, so we have a compound inequality:
[B]125 <= w <= 135[/B]

Brighthouse charges $120 a month for their basic plan, plus $2.99 for each on demand movie you buy.

Brighthouse charges $120 a month for their basic plan, plus $2.99 for each on demand movie you buy. Write and solve and inequality to find how many on demand movies could you buy if you want your bill to be less than $150 for the month.
Let x equal to the number room movie rentals per month. Our inequality is:
120 + 2.99x < 150
To solve for the number of movies, Add 120 to each side
2.99x < 30
Divide each side by 2.99
x < 10.03, which means 10 since you cannot buy a fraction of a movie

Carmen has $30 in store bucks and a 25% discount coupon for a local department store. What maximum d

Carmen has $30 in store bucks and a 25% discount coupon for a local department store. What maximum dollar amount can Carmen purchase so that after her store bucks and discount are applied, her total is no more than $60 before sales tax
Let the original price be p.
p
Apply 25% discount first, which is the same as subtracting 0.25:
p(1 - 0.25)
Subtract 30 for in store buck
p(1 - 0.25) - 30
The phrase [I]no more than[/I] means an inequality using less than or equal to:
p(1 - 0.25) - 30 <= 60
To solve this inequality for p, we [URL='https://www.mathcelebrity.com/1unk.php?num=p%281-0.25%29-30%3C%3D60&pl=Solve']type it in our math engine[/URL] and we get:
[B]p <= 120[/B]

Carmen is serving her child french fries and chicken wings for lunch today. Let f be the number of f

Carmen is serving her child french fries and chicken wings for lunch today. Let f be the number of french fries in the lunch, and let c be the number of chicken wings. Each french fry has 25 calories, and each chicken wing has 100 calories. Carmen wants the total calorie count from the french fries and chicken wings to be less than 500 calories. Using the values and variables given, write an inequality describing this.
We have:
25f + 100c < 50
Note: We use < and not <= because it states less than in the problem.

Chang is serving his child french fries and chicken wings for lunch today. Let f be the number of f

Chang is serving his child french fries and chicken wings for lunch today. Let f be the number of french fries in the lunch, and let c be the number of chicken wings. Each french fry has 25 calories, and each chicken wing has 100 calories. Chang wants the total calorie count from the french fries and chicken wings to be less than 600 calories. Using the values and variables given, write an inequality describing this.
We have [B]25f + 100c < 600[/B] as our inequality.

Charmaine’s fish tank has 16 liters of water in it. she plans to add 6 liters per minute until the t

Charmaine’s fish tank has 16 liters of water in it. she plans to add 6 liters per minute until the tank has at least 58 liters. What are the possible numbers of minutes Charmaine could add water?
This is an algebraic inequality. The phrase [I]at least[/I] means greater than or equal to. So we have:
6m + 16 >= 58 <-- This is our algebraic expression/inequality.
To solve this, [URL='https://www.mathcelebrity.com/1unk.php?num=6m%2B16%3E%3D58&pl=Solve']we type this into our search engine [/URL]and we get:
[B]m >= 7[/B]

Choose the best inequality for this scenario. Casey bought a sandwich and a drink for $3.75. If she

Choose the best inequality for this scenario. Casey bought a sandwich and a drink for $3.75. If she has $6.00 to spend, what is the most she can spend on dessert?
Let dessert spend be d. We have:
d <= $6.00 - $3.75
[B]d <= $2.25[/B]

Craig went bowling with $25 to spend. He rented shoes for $5.25 and paid $4.00 for each game. What w

Craig went bowling with $25 to spend. He rented shoes for $5.25 and paid $4.00 for each game. What was the greatest number of games Craig could have played?
Set up the cost function C(g) where g is the number of games Craig plays:
C(g) = Game fee * number of games (g) + shoe rental fee
C(g) = 4g + 5.25
The problem asks for the maximum number of games Craig can play for $25. So we want an inequality of [I]less than or equal to[/I].
4g + 5.25 <= 25
[URL='https://www.mathcelebrity.com/1unk.php?num=4g%2B5.25%3C%3D25&pl=Solve']Type this inequality into our search engine[/URL], and we get:
g <= 4.9375
We want exact games, so we round this down to [B]4 games[/B].

d squared is greater than or equal to 17

d squared is greater than or equal to 17
d squared means we raise the variable d to the power of 2:
d^2
The phrase [I]greater than or equal to[/I] means an inequality. So we set this up using the >= in relation to 17:
[B]d^2 >= 17[/B]

Dawn has less than $60. She wants to buy 3 sweaters. What price of sweaters can she afford if all th

Dawn has less than $60. She wants to buy 3 sweaters. What price of sweaters can she afford if all the sweaters are the same price? Let s be the price of each sweater. Write this as an inequality.
The phrase [I]less than[/I] means an inequality, so we have the following inequality:
3s < 60
To solve this inequality, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=3s%3C60&pl=Show+Interval+Notation']type it in our math engine[/URL] and we get:
s < [B]20[/B]

Division Equality Property Calculator

Demonstrates the Division Equality Property Calculator
Numerical Properties

Division Property Of Inequality

Demonstrates the Division Property Of Inequality
Numerical Properties

Dylans mother tells Dylan he must spend less time playing electronic games. On the weekends he spend

Dylans mother tells Dylan he must spend less time playing electronic games. On the weekends he spends 9.5 hours playing electronic games. If he plays between 13 and 19 hours each week, how many hours does he play games on weekdays?
Let x equal the number of hours Dylan plays electronic games per week.
[U]Set up our inequality:[/U]
13 <= x <= 19
[U]To see how much he plays during weekdays, subtract off the weekend time[/U]
13 - 9.5 <= x <= 19 - 9.5
[B]3.5 <= x <= 9.5[/B]

Emily is three years older than twice her sister Mary’s age. The sum of their ages is less than 30.

Emily is three years older than twice her sister Mary’s age. The sum of their ages is less than 30. What is the greatest age Mary could be?
Let e = Emily's age and m = Mary's age.
We have the equation e = 2m + 3 and the inequality e + m < 30
Substitute the equation for e into the inequality:
2m + 3 + m < 30
Add the m terms
3m + 3 < 30
Subtract 3 from each side of the inequality
3m < 27
Divide each side of the inequality by 3 to isolate m
m < 9
Therefore, the [B]greatest age[/B] Mary could be is 8, since less than 9 [U]does not include[/U] 9.

Equation and Inequalities

Solves an equation or inequality with 1 unknown variable and no exponents as well as certain absolute value equations and inequalities such as |x|=c and |ax| = c where a and c are constants. Solves square root, cube root, and other root equations in the form ax^2=c, ax^2 + b = c. Also solves radical equations in the form asqrt(bx) = c. Also solves open sentences and it will solve one step problems and two step equations. 2 step equations and one step equations and multi step equations

Express the fact that x differs from 3 by less than 2/7 as an inequality involving absolute value. S

Express the fact that x differs from 3 by less than 2/7 as an inequality involving absolute value. Solve for x.
Let's build this algebraic expression in pieces:
The phrase [I]differs from[/I] means a difference.
x - 3
By less than 2/7 means we use the < sign compared to 2/7
x - 3 < 2/7
Finally, the problem says we involve absolute value. So we write this as:
[B]|x - 3| < 2/7[/B]

Five less than a number is at least -7 and at most 7.

Five less than a number is at least -7 and at most 7.
A number signifies an arbitrary variable, let's call it x.
Five less than a number: x - 5
Is at least -7 means greater than or equal to and at most 7 means less than or equal to, so we have a joint inequality:
[B]-7 <= x - 5 <= 7[/B]

For her phone service, Maya pays a monthly fee of $27 , and she pays an additional $0.04 per minu

For her phone service, Maya pays a monthly fee of $27 , and she pays an additional $0.04 per minute of use. The least she has been charged in a month is $86.04 . What are the possible numbers of minutes she has used her phone in a month? Use m for the number of minutes, and solve your inequality for m .
Maya's cost function is C(m), where m is the number of minutes used.
C(m) = 0.04m + 27
We are given C(m) = $86.04. We want her cost function [I]less than or equal[/I] to this.
0.04m + 27 <= 86.04
[URL='https://www.mathcelebrity.com/1unk.php?num=0.04m%2B27%3C%3D86.04&pl=Solve']Type this inequality into our search engine[/URL], and we get [B]m <= 1476[/B].

Four more then double a number is greater than 2

Four more then double a number is greater than 2
Double a number:
A number implies an arbitrary variable, let's call it "x". Double means multiply this by 2
2x
Four more than this:
2x + 4
Now, we set this expression as an inequality greater than 2
[B]2x + 4 > 2[/B]

Gary is buying chips. Each bag costs $3.50. He has $40 to spend. Write an inequality to represent th

Gary is buying chips. Each bag costs $3.50. He has $40 to spend. Write an inequality to represent the number of chip bags, c, he can afford.
Gary's spend is found by this inequality:
[B]3.50c <= 40
[/B]
To solve this inequality, [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=3.50c%3C%3D40&pl=Show+Interval+Notation']we type it in our search engine[/URL] and we get:
[B]c <= 11.43[/B]

Given: BC = EF AC = EG AB = 10 BC = 3 Prove FG = 10

Given: BC = EF
AC = EG
AB = 10
BC = 3
Prove FG = 10
[LIST]
[*]AC = AB + BC (Segment Addition Postulate)
[*]AB = 10, BC = 3 (Given)
[*]AC = 10 + 3 (Substitution Property of Equality)
[*]AC = 13 (Simplify)
[*]AC = EG, BC = EF (Given)
[*]EG = 13, EF = 3 (Segment Equivalence)
[*]EG = EF + FG (Segment Addition Postulate)
[*]13 = 3 + FG (Substitution Property of Equality)
[*]FG = 10 (Subtraction Property)
[/LIST]

Goal is to take at least 10,000 steps per day. According to your pedometer you have walked 5,274 ste

Goal is to take at least 10,000 steps per day. According to your pedometer you have walked 5,274 steps. Write and solve an inequality to find the possible numbers of steps you can take to reach your goal.
[U]
Subtract off the existing steps (s) from your goal of 10,000[/U]
g >= 10000 - 5274
[B]g >= 4726[/B]
[I]Note: we use >= since 10,000 steps meets the goal as well as anytihng above it[/I]

harley had $500 in his bank account at the beginning of the year. he spends $20 each week on food, c

harley had $500 in his bank account at the beginning of the year. he spends $20 each week on food, clothing, and movie tickets. he wants to have more than $100 at the end of summer to make sure he has enough to purchase some new shoes before school starts. how many weeks, w, can harley withdraw money from his savings account and still have more than $100 to buy new shoes?
Let the number of weeks be w. Harley needs $100 (or more) for shoes. We have the balance in Harley's account as:
500 - 20w >= 100
To solve this inequality for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=500-20w%3E%3D100&pl=Solve']type it in our search engine[/URL] and we get:
[B]w <= 20[/B]

Interval Notation and Set Builder Notation

This calculator translates the following inequality statements to interval notation and set builder notation:

* x < 5

* y <= 5

* z > 5

* a >= 5

* b < 5 or b > 20

* Compound Inequality such as 0 <= c < 4

* |x|<3

* Reverse Interval Notation to Inequality Statement such as (-7,5]

* {x|x<1}

* Word representations of interval notations such as 2 is less than or equal to x is less than or equal to 8

* x < 5

* y <= 5

* z > 5

* a >= 5

* b < 5 or b > 20

* Compound Inequality such as 0 <= c < 4

* |x|<3

* Reverse Interval Notation to Inequality Statement such as (-7,5]

* {x|x<1}

* Word representations of interval notations such as 2 is less than or equal to x is less than or equal to 8

Is it correct to word "10% * 50 + 50" as "10% upper 50"?

Yes, it's close to the upper bound. I just wonder what we interpret the below inequality is?
y > 10% x + x
My two cents: y is greater than 10% upper x
What do you say?

Isabel earns $7.50 per hour on the weekends. Write and solve an inequality to find how many hours sh

Isabel earns $7.50 per hour on the weekends. Write and solve an inequality to find how many hours she needs to work to earn at least $120.
A few things to note:
[LIST]
[*]Earnings = Rate * time
[*]Let h be the number of hours worked
[*]The phrase [I]at least[/I] means greater than or equal to, so we have the following inequality.
[/LIST]
We represent this with the following inequality:
7.5h < 120
To solve this inequality for h, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=7.5h%3C120&pl=Show+Interval+Notation']type it into our math engine[/URL] and we get:
[B]h < 16[/B]

Isabel will run less than 36 minutes today. So far, she has run 22 minutes. What are the possible nu

Isabel will run less than 36 minutes today. So far, she has run 22 minutes. What are the possible numbers of additional minutes she will run?
Set up our inequality. If she ran 22 minutes, we need to find an expression to find out the remaining minutes
x + 22 < 36
Subtract 22 from each side:
x < 14
Remember, she cannot run negative minutes, so our lower bound is 0, so we have:
[B]0 < x < 14
[/B]

jane has 55$ to spend at cedar point. the admission price is 42$ and each soda is 4.25. write an ine

jane has 55$ to spend at cedar point. the admission price is 42$ and each soda is 4.25. write an inequality to show how many sodas he can buy.
Let s be the number of sodas.
Cost for the day is:
Price per soda * s + Admission Price
4.25s + 42
We're told that Jane has 55, which means Jane cannot spend more than 55. Jane can spend up to or less than 55. We write this as an inequality using <= 55
[B]4.25s + 42 <= 55[/B]
[B][/B]
If the problems asks you to solve for s, we type it in our math engine and we get:
Solve for [I]s[/I] in the inequality 4.25s + 42 ? 55
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 42 and 55. To do that, we subtract 42 from both sides
4.25s + 42 - 42 ? 55 - 42
[SIZE=5][B]Step 2: Cancel 42 on the left side:[/B][/SIZE]
4.25s ? 13
[SIZE=5][B]Step 3: Divide each side of the inequality by 4.25[/B][/SIZE]
4.25s/4.25 ? 13/4.25
[B]s ? 3.06[/B]

Jinas final exam has true/false questions, worth 3 points each, and multiple choice questions, worth

Jinas final exam has true/false questions, worth 3 points each, and multiple choice questions, worth 4 points each. Let x be the number of true/false questions she gets correct, and let y be the number of multiple choice questions she gets correct. She needs at least 76 points on the exam to get an A in the class. Using the values and variables given, write an inequality describing this.
At least means greater than or equal to, so we have:
[B]3x + 4y >= 76[/B]

Jody is buying a scrapbook and sheets of designer paper. She has $40 and needs at least $18.25 to bu

Jody is buying a scrapbook and sheets of designer paper. She has $40 and needs at least $18.25 to buy the scrapbook. Each sheet of paper costs $0.34. How many sheets of paper can she buy?
Set up a cost equation for the number of pieces of paper (p):
0.34p + 18.25 <= 40 <-- we have an inequality since we can't go over 40
[URL='https://www.mathcelebrity.com/1unk.php?num=0.34p%2B18.25%3C%3D40&pl=Solve']Type this inequality into our search engine[/URL] and we get:
p <= 63.97
We round down, so we get p = [B]63[/B].

Joey and Romnick play in the same soccer team. Last Saturday, Romnick scored 3 more goals than Joey,

Joey and Romnick play in the same soccer team. Last Saturday, Romnick scored 3 more goals than Joey,but together they scored less than 9 goals. What are the possible number of goal Romnick scored?
Let j be Joey's goals
Let r by Romnick's goals
We're given 1 equation and 1 inequality:
[LIST=1]
[*]r = j + 3
[*]r + j < 9
[/LIST]
Rearranging equation 1 for j, we have:
[LIST=1]
[*]j = r - 3
[*]r + j < 9
[/LIST]
Substitute equation (1) into inequality (2) for j:
r + r - 3 < 9
2r - 3 < 9
[URL='https://www.mathcelebrity.com/1unk.php?num=2r-3%3C9&pl=Solve']Typing this inequality into our math engine[/URL], we get:
[B]r < 6[/B]

Jordan has already scored 153 points this basketball season. If he scores 17 points per game, which

Jordan has already scored 153 points this basketball season. If he scores 17 points per game, which inequality represents the number of addional games he needs to play in order to score at least 255 points for the season?
Let g be the number of games Jordan plays. Total points per game is 17g. And he’s already scored 153. So we need 17g + 153 to be [I]at least [/I]255. The phrase at least means greater than or equal to, so we use the >= operator for our inequality:
[B]17g + 153 >= 255[/B]

Jose has scored 556 points on his math tests so far this semester. To get an A for the semester, he

Jose has scored 556 points on his math tests so far this semester. To get an A for the semester, he must score at least 660 points. Write and solve an inequality to find the minimum number of points he must score on the remaining tests, n, in order to get an A.
We want to know n, such that 556 + n >= 660. <-- We use >= symbol since at least means greater than or equal to.
556 + n >= 660
Use our [URL='http://www.mathcelebrity.com/1unk.php?num=556%2Bn%3E%3D660&pl=Solve']equation/inequality calculator[/URL], we get [B]n >= 104[/B]

JP's age is twice the age of Reyna. The sum of their ages does not exceed 51

JP's age is twice the age of Reyna. The sum of their ages does not exceed 51
Let JP's age be j. Let Reyna's age be r. We're given two expressions:
[LIST=1]
[*]w = 2r
[*]r + w <= 51. ([I]Does not exceed means less than or equal to)[/I]
[/LIST]
We substitute (1) into (2) for w to get the inequality:
r + 2r <= 51
To solve this inequality, we type it in our search engine and we get:
[B]r <= 17[/B]

Juan has d dimes and q quarters in his pocket. The total value of the coins is less than $14.75. Whi

Juan has d dimes and q quarters in his pocket. The total value of the coins is less than $14.75. Which inequality models this situation?
[U]Let d be the number of dimes and q be the number of quarters[/U]
[B]0.1d + 0.25q < 14.75[/B]

Juan has d dimes and q quarters in his pocket. The total value of the coins is less than $14.75. Whi

Juan has d dimes and q quarters in his pocket. The total value of the coins is less than $14.75. Which inequality models this situation?
Since dimes are worth $0.10 and quarters are worth $0.25, we have:
[B]0.10d + 0.25q < 14.75[/B]

Julie has $300 to plan a dance. There is a one-time fee of $75 to reserve a room. It also costs $1.5

Julie has $300 to plan a dance. There is a one-time fee of $75 to reserve a room. It also costs $1.50 per person for food and drinks. What is the maximum number of people that can come to the dance?
Let each person be p. We have the following relationship for cost:
1.50p + 75 <=300
We use the <= sign since we cannot go over the $300 budget.
[URL='https://www.mathcelebrity.com/1unk.php?num=1.50p%2B75%3C%3D300&pl=Solve']We type this inequality into our search engine[/URL], and we get:
p <= 150
Since we have the equal sign within the inequality, the maximum number of people that can come to the dance is [B]150.[/B]

Karen bought a bucket of popcorn at the movies for $5. She also bought some candy for $2 each. Karen

Karen bought a bucket of popcorn at the movies for $5. She also bought some candy for $2 each. Karen has to spend less than $15 on the popcorn and candy. Which inequality can be used to find c, the number of candies that Karen could have bought?
Since the candy cost is the product of price and quantity, we have:
2c + 5 < 15
To solve this inequality for c, we [URL='https://www.mathcelebrity.com/1unk.php?num=2c%2B5%3C15&pl=Solve']type it in our math engine [/URL]and we get:
[B]c < 5[/B]

Kate spent at most $2.50 on apples and oranges. She bought 5 apples at $0.36 each. What is the most

Kate spent at most $2.50 on apples and oranges. She bought 5 apples at $0.36 each. What is the most She spent on oranges
[U]Assumptions and givens:[/U]
[LIST]
[*]Let a be the total cost of apples
[*]Let o be the total cost of oranges
[/LIST]
The phrase [I]at most[/I] means less than or equal to, so we have:
a + o <= 2.50
[U]Find the cost of apples (a)[/U]
a = price per apple * quantity of apples
a = 0.36 * 5
a = 1.8
Our new inequality with a = 1.8 is:
1.8 + o <= 2.50
[URL='https://www.mathcelebrity.com/1unk.php?num=1.8%2Bo%3C%3D2.50&pl=Solve']Typing this inequality into our search engine[/URL], we get:
[B]o <= 0.7[/B]

keisha is babysitting at 8$ per hour to earn money for a car. So far she has saved $1300. The car th

keisha is babysitting at 8$ per hour to earn money for a car. So far she has saved $1300. The car that keisha wants to buy costs at least $5440. How many hours does Keisha need to babysit to earn enough to buy the car
Set up the Earning function E(h) where h is the number of hours Keisha needs to babysit:
E(h) = 8h + 1300
The question asks for h when E(h) is at least 5440. The phrase [I]at least[/I] means an inequality, which is greater than or equal to. So we have:
8h + 1300 >= 5440
To solve this inequality, we [URL='https://www.mathcelebrity.com/1unk.php?num=8h%2B1300%3E%3D5440&pl=Solve']type it in our search engine[/URL] and we get:
h >= [B]517.5[/B]

Keith is going to Renaissance Festival with $120 to pay for his admission, food and the cost of game

Keith is going to Renaissance Festival with $120 to pay for his admission, food and the cost of games. He spends a total of $85 on admission and food. Games cost $5 each. Which inequality models the maximum number of games Keith can play.
Let the number of games be g. Keith can spend less than or equal to 120. So we have
[B]5g + 85 <= 120
[/B]
If we want to solve the inequality for g, we [URL='https://www.mathcelebrity.com/1unk.php?num=5g%2B85%3C%3D120&pl=Solve']type it in our search engine[/URL] and we have:
g <= 7

kim wants to buy candy for 4 dollars a pound. if she wants to spend less than 20 dollars, how many b

kim wants to buy candy for 4 dollars a pound. if she wants to spend less than 20 dollars, how many bags can she buy
Since cost = price * quantity, we have the following inequality with b as the number of bags:
4b < 20
To solve this inequality for b, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=4b%3C20&pl=Show+Interval+Notation']type it in our search engine[/URL] and we get:
[B]b < 5[/B]

kira will spend less than 27 on gifts. so far, she has spent 12$. what are the possible additional a

kira will spend less than 27 on gifts. so far, she has spent 12$. what are the possible additional amounts she will spend?
The key word in this problem is [I]less than[/I]. So we know this is an inequality.
Let s be Kira's possible spend. We have:
s + 12 < 27
To solve for s in this inequality, we subtract 12 from each side:
s + 12 - 12 < 27 - 12
Cancel the 12's on the left side, and we get:
[B]s < 15
[/B]
[I]The summary here is Kira can spend anything up to [U]but not including[/U] 15[/I]

Let p be what Peter earns hourly. Peter earns less than 9 dollars an hour.

p < 9 is our inequality.

Lisa has $150 at most to spend on clothes. She wants to buy a pair of jeans for $58 and will spend t

Lisa has $150 at most to spend on clothes. She wants to buy a pair of jeans for $58 and will spend the rest on t-shirts that cost $14 each.
Let the number of t-shirts be t. Lisa can spend up to, but not more than 150. We have the following inequality:
14t + 58 <= 150
To solve this inequality, we [URL='https://www.mathcelebrity.com/1unk.php?num=14j%2B58%3C%3D150&pl=Solve']type it in our search engine[/URL] and we get:
t <= 6.57
To round to a whole number, we round down to [B]t = 6 [/B]

Lisa wants to rent a boat and spend less than $52. The boat costs $7 per hour, and Lisa has a discou

Lisa wants to rent a boat and spend less than $52. The boat costs $7 per hour, and Lisa has a discount coupon for $4 off. What are the possible numbers of hours Lisa could rent the boat?
Calculate discounted cost:
Discounted cost = Full Cost - Coupon
Discounted cost = 52 - 7
Discounted cost = 45
Since price equals rate * hours (h), and we want the inequality (less than) we have:
7h < 52
Using our [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=7h%3C52&pl=Show+Interval+Notation']inequality calculator,[/URL] we see that:
[B]h < 7.42[/B]

Lucas is offered either 15% or $21 off his total shopping bill. How much would have to be spent to m

Lucas is offered either 15% or $21 off his total shopping bill. How much would have to be spent to make the 15% option the best one?
Let the total bill be b. We have:
0.15b > 21 <-- Since 15% is 0.15
Using our [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=0.15b%3E21&pl=Show+Interval+Notation']inequality calculator[/URL], we get [B]b>140[/B].
So any bill greater than $140 will make the 15% off option the best one, since the discount will be higher than $21.

M decreased by the sum of 13 and the number P is less than 12

M decreased by the sum of 13 and the number P is less than 12
The sum of 13 and the number P
13 + P
M decreased by the sum of 13 and the number P
M - (13 + P)
Less than 12 means we set this entire expression less than 12 as an inequality
[B]M - (13 + P) < 12[/B]

M is the midpoint of AB. Prove AB=2AM

M is the midpoint of AB. Prove AB=2AM
M is the midpoint of AB (Given)
AM = MB (Definition of Congruent Segments)
AM + MB = AB (Segment Addition Postulate)
AM + AM = AB (Substitution Property of Equality)
2AM = AB (Distributive property)

Maggie is shopping for her friend's party. She has a budget of $40 to spend. She needs to get a bann

Maggie is shopping for her friend's party. She has a budget of $40 to spend. She needs to get a banner for $25 and candy necklaces that cost $1.25 each. Write an inequality for the budget.
Let n be the necklaces. Since Maggie can spend [I]up to[/I] $40, we have the following inequality:
[B]1.25n + 25 <=40
[/B]
If you have to solve for n in the inequality, we [URL='https://www.mathcelebrity.com/1unk.php?num=1.25n%2B25%3C%3D40&pl=Solve']type it in our math engine[/URL] and we get:
[B]n < = 12[/B]

Matilda needs at least $112 to buy an new dress. She has already saved $40. She earns $9 an hour bab

Matilda needs at least $112 to buy an new dress. She has already saved $40. She earns $9 an hour babysitting. Write and solve and inequality to find how many hours she will need to babysit to buy the dress.
Subtract remaining amount needed after savings:
112 - 40 = 72
Let h be her hourly wages for babysitting. We have the equation:
[B]9h = 72[/B]
Divide each side by 9
[B]h = 8[/B]

Miguel has $80 in his bank and saves $2 a week. Jesse has $30 in his bank but saves $7 a week. In ho

Miguel has $80 in his bank and saves $2 a week. Jesse has $30 in his bank but saves $7 a week. In how many weeks will Jesse have more in his bank than Miguel?
[U]Set up the Bank value B(w) for Miguel where w is the number of weeks[/U]
B(w) = Savings Per week * w + Current Bank Balance
B(w) = 2w + 80
[U]Set up the Bank value B(w) for Jesse where w is the number of weeks[/U]
B(w) = Savings Per week * w + Current Bank Balance
B(w) = 7w + 30
The problem asks when Jesse's account will be more than Miguel's. So we set up an inequality where:
7w + 30 > 2w + 80
To solve this inequality, we [URL='https://www.mathcelebrity.com/1unk.php?num=7w%2B30%3E2w%2B80&pl=Solve']type it in our search engine[/URL] and we get:
[B]w > 10[/B]

Mr. Smith wants to spend less than $125 at a zoo. A ticket cost $7 he is taking 2 kids with him. Use

Mr. Smith wants to spend less than $125 at a zoo. A ticket cost $7 he is taking 2 kids with him. Use p to represent the other money he can spend there.
2 kids and Mr. Smith = 3 people.
Total Ticket Cost is 3 people * 7 per ticket = 21
If he has 125 to spend, we have the following inequality using less than or equal to (<=) since he can spend up to or less than 125:
p + 21 <= 125
Subtract 21 from each side:
[B]p <= 104[/B]

Multiplication Equality Property

Demonstrates the Multiplication Equality Property
Numerical Properties

Multiplication Property Of Inequality

Demonstrates the Multiplication Property Of Inequality
Numerical Properties

Nine less than a number is no more than 8 and no less than 3

Nine less than a number is no more than 8 and no less than 3
A number is denoted as an arbitrary variable, let's call it x.
We have a double inequality:
[LIST=1]
[*]No more than 8 means less than or equal to 8
[*]No less than 3 means greater than or equal to 3
[/LIST]
[B]3 <= x <= 8[/B]

Nine less than the product of 2 and y is not less than 15

The product of 2 and y means we multiply
2y
Nine less than that product means we subtract 9
2y - 9
Finally, the entire expression is not less than 15, which means 15 or more. We use greater than or equal to
[B]2y - 9 >= 15
[/B]
If you want to solve this inequality and write the interval notation, use [URL='http://www.mathcelebrity.com/1unk.php?num=2y-9%3E%3D15&pl=Solve']our calculator[/URL].

Normal body temperature is 98.6 ? F. Write an inequality that describes the temperature

Normal body temperature is 98.6 ? F. Write an inequality that describes the temperature, T, of people with above normal temperatures.
Above means greater than, so we set up the inequality:
[B]T > 98.6 ?[/B]

Notebooks cost $1.39 each. What are the possible numbers of notebooks that can be purchased with $10

Notebooks cost $1.39 each. What are the possible numbers of notebooks that can be purchased with $10?
Let n be the number of notebooks you can purchase. We have the following inequality:
1.39n <= 10
Divide each side by 1.39
n <= 7.194
We want whole notebooks, we cannot buy fractions of notebooks, so we have:
n <= 7
The question asks for the possible numbers of notebooks we can buy. This implies we buy at least 1, but our inequality says not more than 7. So our number set is:
[B]N = {1, 2, 3, 4, 5, 6, 7}[/B]

People with a drivers license are at least 16 years old and no older than 85 years old

People with a drivers license are at least 16 years old and no older than 85 years old.
Set up the inequality, where p represents the people:
[LIST=1]
[*]The phrase [I]at least[/I] means greater than or equal to. So we use the >= sign. 16 <= p
[*]The phrase [I]no older than[/I] means less than or equal to. So we use the <= sign. p <= 85
[/LIST]
Combine these inequalities, and we get:
[B]16 <= p <= 85[/B]
To see the interval notation for this inequality and all possible values, visit the [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=16%3C%3Dp%3C%3D85&pl=Show+Interval+Notation']interval notation calculator[/URL].

Pet supplies makes a profit of $5.50 per bag, if the store wants to make a profit of no less than $5

Pet supplies makes a profit of $5.50 per bag, if the store wants to make a profit of no less than $5225, how many bags does it need to sell?
5.5ob >= $5,225
Divide each side of the inequality by $5.50
b >=9.5 bags, so round up to a whole number of 10 bags.

rectangle abcd prove: triangle adc is congruent to triangle bcd

rectangle abcd prove: triangle adc is congruent to triangle bcd
1. Given: ABCD is a rectangle
2. AB = CD since opposite sides of rectangle are congruent
3. BC = AD since opposite sides of rectangle are congruent
4. AC = AC by the Reflexive Property of Equality
5. triangle ADC = triangle CBA by the Side-Side-Side (SSS) Property

Rhonda raised $245 for her softball team's fundraiser.She wants to raise no less than $455.Write and

Rhonda raised $245 for her softball team's fundraiser.She wants to raise no less than $455.Write and solve an inequality to determine how much more money Rhonda must raise to reach her goal. Let d represent the amount of money in dollars Rhonda must raise to reach her goal.
The phrase [I]no less than[/I] is an inequality using the greater than or equal sign:
d >= 455 - 245
d >= [B]210[/B]

Rita has spent $16 so far on gifts. The additional amount she will spend will be less than $14. What

Rita has spent $16 so far on gifts. The additional amount she will spend will be less than $14. What are the possible total amounts she will spend?
Rita will spend at least another cent on other gifts above the $16 she spent so far, but no more than $14. Also, the problem says less than 14. 16 + 14 is 30, so that is the top end of her spending.
Let's say her remaining spending is s. Set up the inequality for possible spending values.
[B]16 < s < 30[/B]

Ryan buys candy that costs $4 per pound. He will buy at least 12 pounds of candy. What are the possi

Ryan buys candy that costs $4 per pound. He will buy at least 12 pounds of candy. What are the possible amounts he will spend on candy?
Clue for you: the phrase [I]at least[/I] means an inequality.
Let s be the spend on candy.
Cost = Price * quantity
Cost = 4 * 12
Cost = 48
The phrase [I]at least[/I] means greater than or equal to:
[B]s >= 48[/B]

Sarah has $250 in her account. She withdraws $25 per week. How many weeks can she withdraw money fro

Sarah has $250 in her account. She withdraws $25 per week. How many weeks can she withdraw money from her account and still have money left?
Let w be the number of weeks. We have the following equation for the Balance after w weeks:
B(w) = 250 - 25w [I]we subtract for withdrawals[/I]
The ability to withdrawal money means have a positive or zero balance after withdrawal. So we set up the inequality below:
250 - 25w >= 0
To solve this inequality for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=250-25w%3E%3D0&pl=Solve']type it in our search engine[/URL] and we get:
w <= [B]10
So Sarah can withdrawal for up to 10 weeks[/B]

Sarah makes $9 per hour working at a daycare center and $12 per hour working at a restaurant. Next

Sarah makes $9 per hour working at a daycare center and $12 per hour working at a restaurant. Next week, Sarah is scheduled to work 8 hours at the daycare center. Which of the following inequalities represents the number of hours (h) that Sandra needs to work at the restaurant next week to earn at least $156 from these two jobs?
Set up Sarah's earnings function E(h) where h is the hours Sarah must work at the restaurant:
12h + 9(8) >= 156 <-- The phrase [I]at least[/I] means greater than or equal to, so we set this up as an inequality. Also, the daycare earnings are $9 per hour * 8 hours
Multiplying through and simplifying, we get:
12h + 72 >= 156
We [URL='https://www.mathcelebrity.com/1unk.php?num=12h%2B72%3E%3D156&pl=Solve']type this inequality into the search engine[/URL], and we get:
[B]h>=7[/B]

Six less than twice a number is at least -1 and at most 1

First, the phrase [I]a number[/I] means we choose an arbitrary variable, let's call it x.
Twice a number means we multiply it by 2.
2x
Six less than that means we subtract 6
2x - 6
Now, the last piece, we set up an inequality. At least -1 means greater than or equal to 1. At most 1 means less than or equal to 1. Notice, for both points, we include the number.
-1 <= 2x - 6 <= 1

Subtraction Equality Property

Demonstrates the Subtraction Equality Property
Numerical Properties

Subtraction Property Of Inequality

Demonstrates the Subtraction Property Of Inequality
Numerical Properties

Suppose we need 4 eggs to make a cake. If there are 24 eggs, write an inequality representing the po

Suppose we need 4 eggs to make a cake. If there are 24 eggs, write an inequality representing the possible number of cakes we can make.
Set up a proportion of eggs to cakes where c is the number of cakes per 24 eggs:
4/1 <= 24/c
[URL='https://www.mathcelebrity.com/prop.php?num1=4&num2=24&den1=1&den2=c&propsign=%3C&pl=Calculate+missing+proportion+value']Typing this proportion inequality into our search engine[/URL], we get:
[B]c <= 6[/B]

The auditorium can hold a maximum of 150 people

The auditorium can hold a maximum of 150 people
We want an inequality for the number of people (p) in the auditorium.
The word [I]maximum[/I] means [I]no more than[/I] or [I]less than or equal to[/I]. So we have:
[B]p <= 150[/B]

the average, a, is at least 85

the average, a, is at least 85
At least is an inequality. It also means greater than or equal to, so we have:
[B]a >= 85[/B]

The base of a triangle with a height of 7 units is represented by the formula b=2/7A. The base of th

The base of a triangle with a height of 7 units is represented by the formula b=2/7A. The base of the triangle is less than 10 units. Write and solve an inequality that represents the possible area A of the triangle
We're given:
b=2/7A
We're also told that b is less than 10. So we have:
2/7A < 10
2A/7 < 10
Cross multiply:
2A < 7 * 10
2A < 70
Divide each side of the inequality by 2 to isolate A
2A/2 < 70/2
Cancel the 2's on the left side and we get:
A < [B]35[/B]

The cost of 25 apples is less than $9.50. The cost of 12 apples is more than 3.60. What are the poss

The cost of 25 apples is less than $9.50. The cost of 12 apples is more than 3.60. What are the possible prices of one apple?
Let a be the price of each apple. We're given 2 inequalities:
[LIST=1]
[*]25a < 9.50
[*]12a > 3.60
[/LIST]
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=25a%3C9.50&pl=Show+Interval+Notation']Typing 25a < 9.50 into our search engine[/URL], we get a < 0.38
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=12a%3E3.60&pl=Show+Interval+Notation']Typing 12a > 360 into our search engine[/URL], we get a > 0.3
Therefore, the possible prices a of one apple are expressed as the inequality:
[B]0.3 < a < 0.38[/B]

The cost of a field trip is $220 plus $7 per student. If the school can spend at most $500, how many

The cost of a field trip is $220 plus $7 per student. If the school can spend at most $500, how many students can go on the field trip?
Set up the inequality where s is the number of students:
C(s) = 220 + 7s
We want C(s) <= 500, since at most means no more than
220 + 7s <= 500
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=220%2B7s%3C%3D500&pl=Solve']inequality calculator[/URL], we get:
[B]s <= 40[/B]

The cost of renting a rototiller is $19.50 for the first hour and $7.95 for each additional hour. Ho

The cost of renting a rototiller is $19.50 for the first hour and $7.95 for each additional hour. How long can a person have the rototiller if the cost must be less than $95?
Setup the inequality:
$19.50 + $7.95x < $95
Subtract 19.50 from both sides:
7.95x < 75.50
Divide each side of the inequality by 7.95 to isolate x
x < 9.5
The next lowest integer is 9. So we take 9 + the first hour of renting to get [B]10 total hours[/B].
Check our work:
$7.95 * 9.5 + $19.50
$71.55 + $19.50 = $91.05

The cost to rent a construction crane is 450 per day plus 150 per hour. What is the maximum number o

The cost to rent a construction crane is 450 per day plus 150 per hour. What is the maximum number of hours the crane can be used each day if the rental cost is not to exceed 1650 per day?
Set up the cost function where h is the number of hours:
C(h) = 150h + 450
We want C(h) <= 1650:
150h + 450 <= 1650
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=150h%2B450%3C%3D1650&pl=Solve']equation/inequality solver[/URL], we get:
[B]h <= 8[/B]

The cube of x is less than 15

The cube of x is less than 15
The cube of x means we raise x to the 3rd power:
x^3
Less than 15 means we setup the following inequality
[B]x^3 < 15[/B]

The difference between A and B is no less than 30

The difference between A and B is no less than 30
The difference between means we subtract.
No less than means greater than or equal to, so we have the following inequality;
[B]A - B >= 30[/B]

the difference between A and B is no less than 30.

the difference between A and B is no less than 30.
The difference between a and b:
a - b
The phrase [I]no less than[/I] means an inequality. You can also say this as [I]greater than or equal to[/I].
[B]a - b >= 30[/B]

The difference of twice a number and 4 is at least -27

The difference of twice a number and 4 is at least -27.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
Twice a number means multiply the number by 2
2x
[I]and 4[/I] means we add 4 to our expression:
2x + 4
[I]Is at least[/I] means an inequality. In this case, it's greater than or equal to:
[B]2x + 4 >= -27
[/B]
To solve this inequality, [URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B4%3E%3D-27&pl=Solve']type it in the search engine[/URL].

The difference of twice a number and 6 is at most 28

The difference of twice a number and 6 is at most 28
This is an algebraic expression. Let's take it in parts:
[LIST=1]
[*]The phrase [I]a number[/I], means an arbitrary variable, let's call it x
[*]Twice this number means we multiply x by 2: 2x
[*][I]The difference of[/I] means subtract, so we subtract 6 to 2x: 2x - 6
[*][I]Is at most [/I]means less than or equal to, so we create an inequality where 2x - 6 is less than or equal to 28, using the <= sign
[/LIST]
[B]2x - 6 <= 28
[/B]
If you wish to solve this inequality, [URL='https://www.mathcelebrity.com/1unk.php?num=2x-6%3C%3D28&pl=Solve']click this link[/URL].

the difference of twice a number and 8 is at most -30

the difference of twice a number and 8 is at most -30.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
Twice this number means we multiply by 2, so we have 2x.
We take the difference of 2x and 8, meaning we subtract 8:
2x - 8
Finally, the phrase [I]at most[/I] means an inequality, also known as less than or equal to:
[B]2x - 8 <= 30 <-- This is our algebraic expression
[/B]
To solve this, we [URL='https://www.mathcelebrity.com/1unk.php?num=2x-8%3C%3D30&pl=Solve']type it into the search engine[/URL] and get x <= 19.

The longest bridge in america is 1700 ft long. Write an inequality that describes the length of ever

The longest bridge in america is 1700 ft long. Write an inequality that describes the length of every bridge.
Let the bridge length be b. Since no bridge will ever be greater than 1700 ft, we have:
[B]b <= 1700[/B]

The minimum daily requirement of vitamin C for 14 year olds is at least 50 milligrams per day. An av

The minimum daily requirement of vitamin C for 14 year olds is at least 50 milligrams per day. An average sized apple contains 6 milligrams of vitamin C. How many apples would a person have to eat each day to satisfy this requirement?
Let a be the number of apples required. The phrase [I]at least[/I] means greater than or equal to, so we have the inequality:
6a >= 50
To solve this inequality, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=6a%3E%3D50&pl=Show+Interval+Notation']type it in our math engine[/URL] and we get:
[B]a >= 8.3333 apples or rounded up to a full number, we get 9 apples[/B]

The price of a baseball glove is no more than $38.95

The price of a baseball glove is no more than $38.95.
Let p be the price of the baseball glove. The phrase "no more than" means less than or equal to. Our inequality is:
p <= $38.95

The product of 8 and a number k is greater than 4 and no more than 16

Let's take this by pieces.
The product of 8 and a number k is written as: 8k.
Since it's greater than 4, but not more than 16, we include this in the middle of an inequality statement.
4 < 8k <= 16
Notice no more than has an equal sign, it means less than or equal to.
Greater does not include an equal sign.

The product of a number b and 3 is no less than 12.

The product of a number b and 3 is no less than 12.
A number b is just written as b. So we have:
The product of b and 3 is no less than 12.
take this in parts:
[LIST]
[*]The product of b and 3: 3b
[*]The phrase [I]is no less than[/I] means an inequality, so we have greater than or equal to. We set 3b greater than or equal to 12
[/LIST]
[B]3b >= 12[/B]

The product of x and 7 is not greater than 21

The product of x and 7 is not greater than 21
The product of x and 7:
7x
Is not greater than means less than or equal to, so we have our algebraic expression:
7x <= 21
If you want to solve this inequality and interval notation, use our [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=7x%3C%3D21&pl=Show+Interval+Notation']calculator[/URL].

The recommended daily calcium intake for a 20-year-old is 1,000 milligrams (mg). One cup of milk con

The recommended daily calcium intake for a 20-year-old is 1,000 milligrams (mg). One cup of milk contains 299 mg of calcium and one cup of juice contains 261 mg of calcium. Which of the following inequalities represents the possible number of cups of milk [I]m[/I] and cups of juice [I]j[/I] a 20-year-old could drink in a day to meet or exceed the recommended daily calcium intake from these drinks alone?
Total calcium = Milk calcium + Juice Calcium
Calculate Milk Calcium:
Milk Calcium = 299m where m is the number of cups of milk
Calculate Juice Calcium:
Juice Calcium = 261j where j is the number of cups of juice
The phrase [I]meet or exceed[/I] means greater than or equal to, so we have an inequality, where Total Calcium is greater than or equal to 1000. So we write our inequality as:
Milk calcium + Juice Calcium >= Total Calcium
[B]299m + 261j >= 1000[/B]

The science club charges 4.50 per car at their car wash. Write and solve and inequality to find how

The science club charges 4.50 per car at their car wash. Write and solve and inequality to find how many cars they have to wash to earn at least 300
Let x be the number of cars they wash. Set up our inequality. Note, at least 300 means 300 or greater, so we use greater than or equal to.
[U]Inequality:[/U]
[B]4.50x >= 300
[/B]
[U]So solve for x, divide each side by 4[/U]
[B]x >= 66.67[/B]

the square of the sum of x and y is less than 20

the square of the sum of x and y is less than 20
The sum of x and y means we add y to x:
x + y
the square of the sum of x and y means we raise the term x + y to the 2nd power:
(x + y)^2
The phrase [I]is less than[/I] means an inequality, so we write this as follows:
[B](x + y)^2 < 20[/B]

The sum of 2 and w is less than or equal to 27.

The sum of 2 and w is less than or equal to 27.
Take this algebraic expression in parts:
[LIST]
[*]The sum of 2 and w: 2 + w
[*]The phrase [I]less than or equal to[/I] means an inequality, using the <= sign.
[/LIST]
[B]2 + w <= 27[/B]

The sum of 2x and y is at least 20

The sum of 2x and y is at least 20
The sum of 2x and y:
2x + y
The phrase [I]is at least[/I] means an inequality. We write this as >= or greater than or equal to:
[B]2x + y >= 20[/B]

The sum of 3 consecutive integers is greater than 30.

The sum of 3 consecutive integers is greater than 30.
Let the first consecutive integer be n
The second consecutive integer is n + 1
The third consecutive integer is n + 2
The sum is written as:
n + n + 1 + n + 2
Combine like terms:
(n + n + n) + (1 + 2)
3n + 3
The phrase [I]greater than[/I] means an inequality, which we write as:
[B]3n + 3 > 30[/B]

The sum of 5 and 2x is at most 27

The sum of 5 and 2x is at most 27
The sum of 5 and 2x means we add 2x to 5:
5 + 2x
The phrase [I]at most[/I] means less than or equal to, so we have an inequality where 5 + 2x is less than or equal to 27
[B]5 + 2x <= 27[/B]

the sum of 5 and y is less than or equal to -21

the sum of 5 and y is less than or equal to -21
Take this algebraic expression in parts:
The sum of 5 and y means we add y to 5
5 + y
The phrase [I]less than or equal to[/I] -21 means an inequality. We use the <= sign to relate 5 + y to -21
[B]5 + y <= -21[/B]

The sum of 5x and 2x is at least 70

[I]Is at least [/I]means greater than or equal to:
5x + 2x >= 70
If we combine like terms, we have:
7x >=70
We can further simplify by dividing each side of the inequality by 7
x >=10
If you want the interval notation for that, use the [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=x%3E%3D10&pl=Show+Interval+Notation']interval notation calculator[/URL].

The sum of a number b and 3 is greater than 4 and no more than 16

The sum of a number b and 3 is greater than 4 and no more than 16
The sum of a number b and 3:
b + 3
Greater than 4 and no more than 16 means we have a combo inequality:
[LIST]
[*]Greater than 4 means we use a > sign
[*]No more than 16 means less than or equal to, so <=
[/LIST]
[B]4 < b + 3 <= 16[/B]

the sum of a number times 3 and 30 is less than 17

the sum of a number times 3 and 30 is less than 17
A number is denoted as an arbitrary variable, let's call it x.
x
Times 3 means we multiply x by 3:
3x
The sum of a number times 3 and 30 means we add 30 to 3x above
3x + 30
Is less than 17 means we have an inequality, so we set 3x + 30 less than 17
3x + 30 < 17
To solve for x and see the interval notation, use [URL='http://www.mathcelebrity.com/1unk.php?num=3x%2B30%3C17&pl=Solve']our calculator[/URL]:

The sum of x and y is at most 10

The sum of x and y is at most 10
The sum of x and y:
x + y
Is at most 10 means we have an inequality, at most means 10 or less, so less than or equal to
[B]x + y <= 10[/B]

The width of a rectangle is fixed at 4cm. For what lengths will the area be less than 86 cm^2

The width of a rectangle is fixed at 4cm. For what lengths will the area be less than 86 cm^2
The Area (A) of a rectangle is given by:
A = lw
With an area of [I]less than[/I] 86 and a width of 4, we have the following inequality:
4l < 86
To solve for l, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=4l%3C86&pl=Show+Interval+Notation']type this inequality into our search engine[/URL] and we get:
[B]l < 21.5[/B]

There are 5 pencil-cases on the desk. Each pencil-case contains at least 10 pencils, but not more th

[SIZE=4]There are 5 pencil-cases on the desk. Each pencil-case contains at least 10 pencils, but not more than 14 pencils. Which of the following could be the total number of pencils in all 5 cases?
A) 35
B) 45
C) 65
D) 75
[U]Determine the minimum amount of pencils (At least means greater than or equal to):[/U]
Minimum Amount of pencils = Cases * Min Quantity
Minimum Amount of pencils = 5 * 10
Minimum Amount of pencils = 50
[SIZE=4][U]Determine the maximum amount of pencils (Not more than means less than or equal to):[/U]
Maximum Amount of pencils = Cases * Min Quantity
Maximum Amount of pencils = 5 * 14
Maximum Amount of pencils = 70[/SIZE]
So our range of pencils (p) is:
50 <= p <= 70
Now take a look at our answer choices. The only answer which fits in this inequality range is [B]C, 65[/B].
[B][/B][/SIZE]

Three more than 2x is greater than or equal to 1 and less than or equal to 11

This is a double inequality. Let's take it by pieces:
Three more than 2x is denoted as 2x + 3. We add since we see the phrase, [I]more than[/I].
Because it's greater than or equal to 1, we have:
1 <= 2x + 3
Finally, that same phrase is [U]also[/U] less than or equal to 11.
2x + 3 <= 11.
Piecing these two inequalities together, we have:
1 <= 2x + 3 <= 11.

Three more than 2x is greater than or equal to 1 and less than or equal to 11

This is a two-part inequality. Let's take it piece by piece.
Three more than 2x means we add.
2x + 3
It's greater than or equal to 1, denoted below:
1 <= 2x + 3
It's also less than or equal to 11, denoted below
2x + 3 <= 11
Piece these two inequalities together:
1 <= 2x + 3 <= 11

Today is my birthday! Four-fifths of my current age is greater than three-quarters of my age one yea

Today is my birthday! Four-fifths of my current age is greater than three-quarters of my age one year from now. Given that my age is an integer number of years, what is the smallest my age could be?
Let my current age be a. We're given:
4/5a > 3/4(a + 1)
Multiply through on the right side:
4a/5 > 3a/4 + 3/4
Let's remove fractions by multiply through by 5:
5(4a/5) > 5(3a/4) + 5(3/4)
4a > 15a/4 + 15/4
Now let's remove the other fractions by multiply through by 4:
4(4a) > 4(15a/4) + 4(15/4)
16a > 15a + 15
[URL='https://www.mathcelebrity.com/1unk.php?num=16a%3E15a%2B15&pl=Solve']Typing this inequality into our search engine[/URL], we get:
a > 15
This means the smallest [I]integer age[/I] which the problem asks for is:
15 + 1 = [B]16[/B]

Tom is the deli manager at a grocery store. He needs to schedule employees to staff the deli departm

Tom is the deli manager at a grocery store. He needs to schedule employees to staff the deli department at least 260 person-hours per week. Tom has one part-time employeewho works 20 hours per week. Each full-time employee works 40 hours per week. Write an inequality to determine n, the number of full-time employees Tom must schedule, so that his employees will work at least 260 person-hours per week.
Set up the inequality:
[LIST]
[*]Add the part-timer's hours of 20
[*]Full time hours is 40 times n employees
[*]At least means greater than or equal to, so we use the >= sign
[/LIST]
[B]40n + 20 >= 260[/B]

Transitive Property of Equality

Demonstrates the Transitive property of equality using a number.
Numerical Properties

Translate the sentence into an inequality. Twice y is less than 21.

Translate the sentence into an inequality. Twice y is less than 21.
Twice y
2y
Is less than 21 means we have an inequality:
[B]2y < 21[/B]

Translate to an inequality. The cost is smaller than $94,000

Translate to an inequality. The cost is smaller than $94,000.
Let the cost be c. We have:
[B]c<94,000[/B]

Triangle Inequality

This calculator displays 2 scenarios

1) Enter 3 sides of a triangle, and it will determine if the side lengths satisfy the properties of the triangle inequality and form a triangle

2) Enter 2 sides of a triangle, and this will determine an acceptable range for the length of the 3rd side of a triangle so that the 3rd side respects the Triangle Inequality.

1) Enter 3 sides of a triangle, and it will determine if the side lengths satisfy the properties of the triangle inequality and form a triangle

2) Enter 2 sides of a triangle, and this will determine an acceptable range for the length of the 3rd side of a triangle so that the 3rd side respects the Triangle Inequality.

two thirds of a number is no more than -10

two thirds of a number is no more than -10
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Two thirds of a number mean we multiply x by 2/3:
2x/3
The phrase [I]no more than[/I] -10 means less than or equal to -10, so we have an inequality:
[B]2x/3 <= -10[/B]

Tyrese’s sister is 41 inches tall. A ride at the amusement park states that riders must be at least

Tyrese’s sister is 41 inches tall. A ride at the amusement park states that riders must be at least 52 inches tall to ride. Which statements describe how much taller Tyrese’s sister must be to ride?
Let h be the required additional height.
The phrase [I]at least[/I] means an inequality, using the >= sign, so we have:
h + 41 >= 52
If we want another way to express this, we [URL='https://www.mathcelebrity.com/1unk.php?num=h%2B41%3E%3D52&pl=Solve']type this inequality into our math engine[/URL] and we get:
[B]h >= 11[/B]

Vectors

Given 2 vectors A and B, this calculates:

* Length (magnitude) of A = ||A||

* Length (magnitude) of B = ||B||

* Sum of A and B = A + B (addition)

* Difference of A and B = A - B (subtraction)

* Dot Product of vectors A and B = A x B

A ÷ B (division)

* Distance between A and B = AB

* Angle between A and B = θ

* Unit Vector U of A.

* Determines the relationship between A and B to see if they are orthogonal (perpendicular), same direction, or parallel (includes parallel planes).

* Cauchy-Schwarz Inequality

* The orthogonal projection of A on to B, proj_{B}A and and the vector component of A orthogonal to B → A - proj_{B}A

Also calculates the horizontal component and vertical component of a 2-D vector.

* Length (magnitude) of A = ||A||

* Length (magnitude) of B = ||B||

* Sum of A and B = A + B (addition)

* Difference of A and B = A - B (subtraction)

* Dot Product of vectors A and B = A x B

A ÷ B (division)

* Distance between A and B = AB

* Angle between A and B = θ

* Unit Vector U of A.

* Determines the relationship between A and B to see if they are orthogonal (perpendicular), same direction, or parallel (includes parallel planes).

* Cauchy-Schwarz Inequality

* The orthogonal projection of A on to B, proj

Also calculates the horizontal component and vertical component of a 2-D vector.

Victoria is 4 years older than her neighbor. The sum of their ages is no more than 14 years.

Victoria is 4 years older than her neighbor. The sum of their ages is no more than 14 years.
Let Victoria's age be v. And her neighbor's age be n. We're given:
[LIST=1]
[*]v = n + 4
[*]v + n <=14 <-- no more than means less than or equal to
[/LIST]
Substitute Equation (1) into Inequality (2):
(n + 4) + n <= 14
Combine like terms:
2n + 4 <= 14
[URL='https://www.mathcelebrity.com/1unk.php?num=2n%2B4%3C%3D14&pl=Solve']Typing this inequality into our search engine[/URL], we get:
n <= 5
Substituting this into inequality (2):
v + 5 <= 14
[URL='https://www.mathcelebrity.com/1unk.php?num=v%2B5%3C%3D14&pl=Solve']Typing this inequality into our search engine[/URL], we get:
[B]v <= 9[/B]

Video store movie rental plans. Plan A 25 membership fee plus 1.25 for movie. Plan B 40 for unlimite

Video store movie rental plans. Plan A 25 membership fee plus 1.25 for movie. Plan B 40 for unlimited rentals. What number of movies rentals is plan B less than plan A?
Let x equal the number of movies rented and C the cost for rentals
Plan A: C = 1.25x + 25
Plan B: C = 40
Set up the inequality:
1.25x + 25 > 40
Subtract 25 from each side:
1.25x > 15
Divide each side of the inequality by 1.25
x > 12
So [B]13[/B] rentals or more make Plan B less than Plan A.

What pair of consecutive integers gives the following: 7 times the smaller is less than 6 times the

What pair of consecutive integers gives the following: 7 times the smaller is less than 6 times the larger?
Let x and y be consecutive integers, where y = x + 1
We have 7x < 6y as our inequality.
Substituting x, y = x + 1, we have:
7x < 6(x + 1)
7x < 6x + 6
Subtracting x from each side, we have:
x < 6, so y = 6 + 1 = 7
(x, y) = (6, 7)

X is the speed limit is a maximum 65 mph

X is the speed limit is a maximum 65 mph
A maximum of means less than or equal to. Or, no more than. So we have the inequality:
[B]X <= 65[/B]

Yolanda wants to rent a boat and spend less than $41. The boat costs $8 per hour, and Yolanda has a

Yolanda wants to rent a boat and spend less than $41. The boat costs $8 per hour, and Yolanda has a discount coupon for $7 off. What are the possible numbers of hours Yolanda could rent the boat?
A few things to build this problem:
[LIST=1]
[*]Discount subtracts from our total
[*]Cost = Hourly rate * hours
[*]Less than means an inequality using the < sign
[/LIST]
Our inequality is:
8h - 7 < 41
To solve this inequality for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=8h-7%3C41&pl=Solve']type it in our math engine[/URL] and we get:
h < [B]6[/B]

You can spend at most $35. If you buy 5 tickets, how much can you spend on each ticket

You can spend at most $35. If you buy 5 tickets, how much can you spend on each ticket
We're given the number of tickets as 5.
We know cost = price * quantity
Let p = price
The phrase [B]at most[/B] means less than or equal to, so we have:
5p <= 35
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=5p%3C%3D35&pl=Show+Interval+Notation']Plugging this inequality into our search engine[/URL], we have:
[B]p <= 7[/B]

You have $10.00 to spend on tacos. Each taco costs $0.50. Write and solve an inequality that explain

You have $10.00 to spend on tacos. Each taco costs $0.50. Write and solve an inequality that explains how many tacos you can buy.
Let's start with t as the number of tacos.
We know that cost = price * quantity, so we have the following inequality for our taco spend:
[B]0.5t <= 10
[/B]
Divide each side of the inequality by 0.5 to isolate t:
0.5t/0.5 <= 10/0.5
Cancel the 0.5 on the left side and we get:
t <= [B]20
[MEDIA=youtube]yy51EsGi1nM[/MEDIA][/B]

You have $20 to spend on taxi fare. The ride costs $5 plus $2.50 per kilometer. Write the inequality

You have $20 to spend on taxi fare. The ride costs $5 plus $2.50 per kilometer. Write the inequality.
Let k be the number of kilometers. We want our total to be $20 [I]or less. [/I]We have the following inequality:
[B]2.50k + 5 <= 20[/B]

You have $6.50 to make copies. It cost $0.45. Write and solve an equality that represents the number

You have $6.50 to make copies. It cost $0.45. Write and solve an equality that represents the number of copies
Hoow many exact copies can you make? Let the number of copies be c. We have:
0.45c = 6.50
[URL='https://www.mathcelebrity.com/1unk.php?num=0.45c%3D6.50&pl=Solve']Type this equation into our search engine[/URL] and we get:
c = 14.444
We round down and say we can make 14 copies.
[B]c = 14[/B]
Now, if the problem asks you for an [I]inequality[/I], we want to see how many copies we can make without exceeding our $6.50 spend. So it's less than or equal to:
[B]c <= 14[/B]

You have $80. Jeans cost $29 and shirts cost $12. Mom told you to buy one pair of jeans and use the

You have $80. Jeans cost $29 and shirts cost $12. Mom told you to buy one pair of jeans and use the rest of the money to buy shirts. Find the inequality.
Let j be the number of jeans. Let s be the number of shirts. We are given:
[LIST]
[*]Mom told you to buy one pair of jeans. So we have $80 to start with - $29 for 1 pair of jeans = $51 left over
[/LIST]
Now, since shirts cost $12 each, and our total number of shirts we can buy is s, our inequality is [B]12s <= 51[/B].
We want to find the s that makes this inequality true.
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=12s%3C%3D51&pl=Show+Interval+Notation']Run this statement through our calculator[/URL], and we get s <= 4.25. But, we need s to be an integer, so we have s <= 4.

You have to pay 29 a month until you reach 850 how many months will that take

You have to pay 29 a month until you reach 850 how many months will that take.
Let m be the number of months. We set up the inequality:
29m > = 850 <-- We want to know when we meet or exceed 850, so we use greater than or equal to
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=29m%3E%3D850&pl=Show+Interval+Notation']Type this inequality into our search engine[/URL], and we get:
m >= 29.31
We round up to the next integer month, to get [B]m = 30[/B].

you must be 65 or older to join inequality

you must be 65 or older to join inequality
Let a be the age. 65 or older means greater than or equal to 65:
[B]a >=65[/B]

Your brother has $2000 saved for a vacation. His airplane ticket is $637. Write and solve an inequal

Your brother has $2000 saved for a vacation. His airplane ticket is $637. Write and solve an inequality to find how much he can spend for everything else.
Let x be the amount your brother can spend. Subtracting the cost of the plane ticket from savings, we have:
x <= 2000 - 637
[B]x <= 1,363[/B]

“The shortest person in a class is 55 inches tall and tallest person in that class is 75 inches tall

The shortest person in a class is 55 inches tall and tallest person in that class is 75 inches tall. write an absolute value equation that requires the minimum and maximum height. Use X to represent heights.
We write our inequality as:
[B]55 <= X <= 75[/B]