notation - An expression made up of symbols for representing operations, unspecified numbers, relations and any other mathematical objects

-2 <= x +4 < 9

-2 <= x +4 < 9
Subtract 4 from each piece:
-2 - 4 <= x < 5
Simplify:
[B]-6 <= x < 5
[/B]
To find the interval notation, we set up our notation:
[LIST]
[*]The left side has a solid bracket, since we have an equal sign:
[*]The right side has an open parentheses, since we have no equal sign
[*][B][-6, 5)[/B]
[/LIST]

-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

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]

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 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 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 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]

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 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]

Base Change Conversions

Converts a positive integer to Binary-Octal-Hexadecimal Notation or Binary-Octal-Hexadecimal Notation to a positive integer. Also converts any positive integer in base 10 to another positive integer base (Change Base Rule or Base Change Rule or Base Conversion)

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]

brenna wants an easier way to write 0.000000000000000000587

brenna wants an easier way to write 0.000000000000000000587
Using our [URL='https://www.mathcelebrity.com/scinot.php?num=0.000000000000000000587&pl=Calculate']scientific notation calculator[/URL], we get:
[B]5.87 x 10^-19[/B]

Complex Number Operations

Given two numbers in complex number notation, this calculator:

1) Adds (complex number addition), Subtracts (complex number subtraction), Multiplies (complex number multiplication), or Divides (complex number division) any 2 complex numbers in the form a + bi and c + di where i = √-1.

2) Determines the Square Root of a complex number denoted as √a + bi

3) Absolute Value of a Complex Number |a + bi|

4) Conjugate of a complex number a + bi

1) Adds (complex number addition), Subtracts (complex number subtraction), Multiplies (complex number multiplication), or Divides (complex number division) any 2 complex numbers in the form a + bi and c + di where i = √-1.

2) Determines the Square Root of a complex number denoted as √a + bi

3) Absolute Value of a Complex Number |a + bi|

4) Conjugate of a complex number a + bi

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]

Expanded Notation

Writes the expanded notation for a number.

Expected Value

This lesson walks you through what expected value is, expected value notation, the expected value of a discrete random variable, the expected value of a continuous random variable, and expected value properties.

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]

If set A ={1,2,3,4} and B={2,4,6,8}, what is A intersect B

If set A ={1,2,3,4} and B={2,4,6,8}, what is A intersect B
Using our [URL='https://www.mathcelebrity.com/setnotation.php?num1=1%2C2%2C3%2C4&num2=2%2C4%2C6%2C8&pl=Set+Notation+Items']set notation calculator[/URL], we get:
A intersect B = [B]{2, 4}[/B]

In the year 2000, the population of Rahway, New Jersey, was 26500. Express this number in scientific

In the year 2000, the population of Rahway, New Jersey, was 26500. Express this number in scientific notation
26,500 in [URL='https://www.mathcelebrity.com/scinot.php?num=26500&pl=Convert+to+Number']scientific notation is found using our scientific notation calculator[/URL]:
[B]2.65 x 10^4[/B]

Index Form

Writes a number using index form notation

Integers

This lesson walks you through what integers are, how to write integers, integer notation, and what's included in integers

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

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]

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]

L'Hôpital's Rule

This lesson walks you through L'Hôpitals's Rule including the definition, pronunciation, notation, and examples

Lagrange Four Square Theorem (Bachet Conjecture)

Builds the Lagrange Theorem Notation (Bachet Conjecture) for any natural number using the Sum of four squares.

Let U be the set of all integers between ?3 and 3 (including ?3 and 3). Let A={?2,0,1,3}. Find Ac. G

Let U be the set of all integers between ?3 and 3 (including ?3 and 3). Let A={?2,0,1,3}. Find Ac. Give your answer in standard set notation
Ac is anything not in A, but in U. So we have:
Ac = [B]{-3, -1, 2}[/B]

Limit of a Function

This lesson walks you through what limit is, how to write limit notation, and limit theorems

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.

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].

One and one third less x

One and one-third can be written as 4/3.
Less x means minus x, or subtract x.
4/3 - x
Or in mixed number notation:
1 & 1/3 - x

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].

Plane Geometry Operations

Evaluates and simplifies various plane geometry notation and operations

Rational Numbers

This lesson walks you through what rational numbers are, how to write rational numbers, rational number notation, and what's included in rational numbers

Real Numbers

This lesson walks you through what real numbers are, how to write real numbers, real numbers notation, and what's included in real numbers

Roster Notation

Given a set of numbers, this displays the roster notation

ROT-13 Conversions

This calculator converts ROT-13 (rot13) (rotate by 13 places) notation to standard notation or standard notation to ROT-13 notation.

Scientific Notation

* Converts a number into scientific notation and determines order of magnitude

* converts scientific notation to a number (standard notation). Also handles scientific notation operations.

* converts scientific notation to a number (standard notation). Also handles scientific notation operations.

set of all letters in Australia

set of all letters in Australia
We remove duplicate "a's" and treat A and a as the same letters. Our set S is:
S = [B]{a, i, l, r, s, t, u}[/B]
If we want to find the properties of this set, we visit our [URL='https://www.mathcelebrity.com/powerset.php?num=%7Ba%2Ci%2Cl%2Cr%2Cs%2Ct%2Cu%7D&pl=Show+Power+Set']set notation calculator[/URL].

Set Theory Notation

Evaluates and describes various set theory notation

Standard Notation

Calculates the following:

* The numeric notation of word notation

* Standard notation of expanded notation

* The numeric notation of word notation

* Standard notation of expanded notation

Standard Notation

Displays the Standard notation of expanded notation

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 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 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 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 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 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]

Word Notation

Calculates the following:

* The word notation of a number of numeric expression

* The word notation of a number of numeric expression

Write in set builder form {all possible numbers formed by any two of the digits 1 2 5}

Write in set builder form {all possible numbers formed by any two of the digits 1 2 5}
With 3 numbers, we got [URL='https://www.mathcelebrity.com/factorial.php?num=3!&pl=Calculate+factorial']3! = 6[/URL] possible numbers formed by the two digits
[LIST=1]
[*]12
[*]15
[*]21
[*]25
[*]51
[*]52
[/LIST]
In set builder notation, we write this as:
{x : x ? {12, 15, 21, 25, 51, 52})
x such that x is a element of the set {12, 15, 21, 25, 51, 52}

Write the interval (2,5) in set builder notation

Write the interval (2,5) in set builder notation
It's a closed interval, so [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=%5B2%2C5%5D&pl=Show+Interval+Notation']we type in [2,5] into the search engine[/URL], and we get:
[B]{x|2<= x <= 5}[/B]

X is a natural number greater than 6

I saw this ticket come through today.
The answer is x > 6.
Natural numbers are positive numbers not 0. So 1, 2, 3, ...
Let me build this shortcut into the calculator.
Also, here is the[URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=x%3E6&pl=Show+Interval+Notation'] interval notation[/URL] for that expression.

x is smaller than 9 and greater than 4

x is smaller than 9 and greater than 4
Written as:
4 < x < 9
To display the interval notation, use our [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=4%3Cx%3C9&pl=Show+Interval+Notation']interval notation calculator[/URL].

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 $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].