## Solving Inequalities

Sometimes we need to solve Inequalities like these:

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

We call that "solved".

## Example: x + 2 > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − 2

x > 10

## How to Solve

Solving inequalities is very like solving equations , we do most of the same things ...

... but we must also pay attention to the direction of the inequality .

Some things can change the direction !

< becomes >

> becomes <

≤ becomes ≥

≥ becomes ≤

## Safe Things To Do

These things do not affect the direction of the inequality:

- Add (or subtract) a number from both sides
- Multiply (or divide) both sides by a positive number
- Simplify a side

## Example: 3x < 7+3

We can simplify 7+3 without affecting the inequality:

But these things do change the direction of the inequality ("<" becomes ">" for example):

- Multiply (or divide) both sides by a negative number
- Swapping left and right hand sides

## Example: 2y+7 < 12

When we swap the left and right hand sides, we must also change the direction of the inequality :

12 > 2y+7

Here are the details:

## Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra ), like this:

## Example: x + 3 < 7

If we subtract 3 from both sides, we get:

x + 3 − 3 < 7 − 3

And that is our solution: x < 4

In other words, x can be any value less than 4.

## What did we do?

And that works well for adding and subtracting , because if we add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

## What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

## Example: 12 < x + 5

If we subtract 5 from both sides, we get:

12 − 5 < x + 5 − 5

That is a solution!

But it is normal to put "x" on the left hand side ...

... so let us flip sides (and the inequality sign!):

Do you see how the inequality sign still "points at" the smaller value (7) ?

And that is our solution: x > 7

Note: "x" can be on the right, but people usually like to see it on the left hand side.

## Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying ).

But we need to be a bit more careful (as you will see).

## Positive Values

Everything is fine if we want to multiply or divide by a positive number :

## Example: 3y < 15

If we divide both sides by 3 we get:

3y /3 < 15 /3

And that is our solution: y < 5

## Negative Values

Well, just look at the number line!

For example, from 3 to 7 is an increase , but from −3 to −7 is a decrease.

See how the inequality sign reverses (from < to >) ?

Let us try an example:

## Example: −2y < −8

Let us divide both sides by −2 ... and reverse the inequality !

−2y < −8

−2y /−2 > −8 /−2

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.)

So, just remember:

When multiplying or dividing by a negative number, reverse the inequality

## Multiplying or Dividing by Variables

Here is another (tricky!) example:

## Example: bx < 3b

It seems easy just to divide both sides by b , which gives us:

... but wait ... if b is negative we need to reverse the inequality like this:

But we don't know if b is positive or negative, so we can't answer this one !

To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b :

- if b is 1 , then the answer is x < 3
- but if b is −1 , then we are solving −x < −3 , and the answer is x > 3

The answer could be x < 3 or x > 3 and we can't choose because we don't know b .

Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

## A Bigger Example

Example: x−3 2 < −5.

First, let us clear out the "/2" by multiplying both sides by 2.

Because we are multiplying by a positive number, the inequalities will not change.

x−3 2 ×2 < −5 ×2

x−3 < −10

Now add 3 to both sides:

x−3 + 3 < −10 + 3

And that is our solution: x < −7

## Two Inequalities At Once!

How do we solve something with two inequalities at once?

## Solver Title

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- -x+3\gt 2x+1
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- There are four types of inequalities: greater than, less than, greater than or equal to, and less than or equal to.
- What is a inequality in math?
- In math, inequality represents the relative size or order of two values.
- How do you solve inequalities?
- To solve inequalities, isolate the variable on one side of the inequality, If you multiply or divide both sides by a negative number, flip the direction of the inequality.
- What are the 2 rules of inequalities?
- The two rules of inequalities are: If the same quantity is added to or subtracted from both sides of an inequality, the inequality remains true. If both sides of an inequality are multiplied or divided by the same positive quantity, the inequality remains true. If we multiply or divide both sides of an inequality by the same negative number, we must flip the direction of the inequality to maintain its truth.

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- High School Math Solutions – Inequalities Calculator, Rational Inequalities Last post, we talked about solving quadratic inequalities. In this post, we will talk about rational inequalities.... Read More

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- 2.5 Solve Linear Inequalities
- Introduction
- 1.1 Use the Language of Algebra
- 1.2 Integers
- 1.3 Fractions
- 1.4 Decimals
- 1.5 Properties of Real Numbers
- Key Concepts
- Review Exercises
- Practice Test
- 2.1 Use a General Strategy to Solve Linear Equations
- 2.2 Use a Problem Solving Strategy
- 2.3 Solve a Formula for a Specific Variable
- 2.4 Solve Mixture and Uniform Motion Applications
- 2.6 Solve Compound Inequalities
- 2.7 Solve Absolute Value Inequalities
- 3.1 Graph Linear Equations in Two Variables
- 3.2 Slope of a Line
- 3.3 Find the Equation of a Line
- 3.4 Graph Linear Inequalities in Two Variables
- 3.5 Relations and Functions
- 3.6 Graphs of Functions
- 4.1 Solve Systems of Linear Equations with Two Variables
- 4.2 Solve Applications with Systems of Equations
- 4.3 Solve Mixture Applications with Systems of Equations
- 4.4 Solve Systems of Equations with Three Variables
- 4.5 Solve Systems of Equations Using Matrices
- 4.6 Solve Systems of Equations Using Determinants
- 4.7 Graphing Systems of Linear Inequalities
- 5.1 Add and Subtract Polynomials
- 5.2 Properties of Exponents and Scientific Notation
- 5.3 Multiply Polynomials
- 5.4 Dividing Polynomials
- Introduction to Factoring
- 6.1 Greatest Common Factor and Factor by Grouping
- 6.2 Factor Trinomials
- 6.3 Factor Special Products
- 6.4 General Strategy for Factoring Polynomials
- 6.5 Polynomial Equations
- 7.1 Multiply and Divide Rational Expressions
- 7.2 Add and Subtract Rational Expressions
- 7.3 Simplify Complex Rational Expressions
- 7.4 Solve Rational Equations
- 7.5 Solve Applications with Rational Equations
- 7.6 Solve Rational Inequalities
- 8.1 Simplify Expressions with Roots
- 8.2 Simplify Radical Expressions
- 8.3 Simplify Rational Exponents
- 8.4 Add, Subtract, and Multiply Radical Expressions
- 8.5 Divide Radical Expressions
- 8.6 Solve Radical Equations
- 8.7 Use Radicals in Functions
- 8.8 Use the Complex Number System
- 9.1 Solve Quadratic Equations Using the Square Root Property
- 9.2 Solve Quadratic Equations by Completing the Square
- 9.3 Solve Quadratic Equations Using the Quadratic Formula
- 9.4 Solve Equations in Quadratic Form
- 9.5 Solve Applications of Quadratic Equations
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- 12.4 Binomial Theorem

## Learning Objectives

By the end of this section, you will be able to:

- Graph inequalities on the number line
- Solve linear inequalities
- Translate words to an inequality and solve
- Solve applications with linear inequalities

## Be Prepared 2.13

Before you get started, take this readiness quiz.

Translate from algebra to English: 15 > x . 15 > x . If you missed this problem, review Example 1.3 .

## Be Prepared 2.14

Translate to an algebraic expression: 15 is less than x . If you missed this problem, review Example 1.8 .

Graph Inequalities on the Number Line

What number would make the inequality x > 3 x > 3 true? Are you thinking, “ x could be four”? That’s correct, but x could be 6, too, or 37, or even 3.001. Any number greater than three is a solution to the inequality x > 3 . x > 3 .

We show all the solutions to the inequality x > 3 x > 3 on the number line by shading in all the numbers to the right of three, to show that all numbers greater than three are solutions. Because the number three itself is not a solution, we put an open parenthesis at three.

We can also represent inequalities using interval notation . There is no upper end to the solution to this inequality. In interval notation, we express x > 3 x > 3 as ( 3 , ∞ ) . ( 3 , ∞ ) . The symbol ∞ ∞ is read as “ infinity .” It is not an actual number.

Figure 2.2 shows both the number line and the interval notation.

We use the left parenthesis symbol, (, to show that the endpoint of the inequality is not included. The left bracket symbol, [, shows that the endpoint is included.

The inequality x ≤ 1 x ≤ 1 means all numbers less than or equal to one. Here we need to show that one is a solution, too. We do that by putting a bracket at x = 1 . x = 1 . We then shade in all the numbers to the left of one, to show that all numbers less than one are solutions. See Figure 2.3 .

There is no lower end to those numbers. We write x ≤ 1 x ≤ 1 in interval notation as ( − ∞ , 1 ] . ( − ∞ , 1 ] . The symbol − ∞ − ∞ is read as “negative infinity.” Figure 2.3 shows both the number line and interval notation.

## Inequalities, Number Lines, and Interval Notation

The notation for inequalities on a number line and in interval notation use the same symbols to express the endpoints of intervals.

## Example 2.48

Graph each inequality on the number line and write in interval notation.

ⓐ x ≥ −3 x ≥ −3 ⓑ x < 2.5 x < 2.5 ⓒ x ≤ − 3 5 x ≤ − 3 5

## Try It 2.95

Graph each inequality on the number line and write in interval notation: ⓐ x > 2 x > 2 ⓑ x ≤ −1.5 x ≤ −1.5 ⓒ x ≥ 3 4 . x ≥ 3 4 .

## Try It 2.96

Graph each inequality on the number line and write in interval notation: ⓐ x ≤ −4 x ≤ −4 ⓑ x ≥ 0.5 x ≥ 0.5 ⓒ x < − 2 3 . x < − 2 3 .

What numbers are greater than two but less than five? Are you thinking say, 2.5 , 3 , 3 2 3 , 4 , 4.99 2.5 , 3 , 3 2 3 , 4 , 4.99 ? We can represent all the numbers between two and five with the inequality 2 < x < 5 . 2 < x < 5 . We can show 2 < x < 5 2 < x < 5 on the number line by shading all the numbers between two and five. Again, we use the parentheses to show the numbers two and five are not included. See Figure 2.4 .

## Example 2.49

ⓐ −3 < x < 4 −3 < x < 4 ⓑ −6 ≤ x < −1 −6 ≤ x < −1 ⓒ 0 ≤ x ≤ 2.5 0 ≤ x ≤ 2.5

## Try It 2.97

Graph each inequality on the number line and write in interval notation:

ⓐ −2 < x < 1 −2 < x < 1 ⓑ −5 ≤ x < −4 −5 ≤ x < −4 ⓒ 1 ≤ x ≤ 4.25 1 ≤ x ≤ 4.25

## Try It 2.98

ⓐ −6 < x < 2 −6 < x < 2 ⓑ −3 ≤ x < −1 −3 ≤ x < −1 ⓒ 2.5 ≤ x ≤ 6 2.5 ≤ x ≤ 6

Solve Linear Inequalities

A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. A linear inequality is an inequality in one variable that can be written in one of the forms, a x + b < c , a x + b < c , a x + b ≤ c , a x + b ≤ c , a x + b > c , a x + b > c , or a x + b ≥ c . a x + b ≥ c .

## Linear Inequality

A linear inequality is an inequality in one variable that can be written in one of the following forms where a , b , and c are real numbers and a ≠ 0 a ≠ 0 :

When we solved linear equations, we were able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities.

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example:

Notice that the inequality sign stayed the same.

This leads us to the Addition and Subtraction Properties of Inequality.

## Addition and Subtraction Property of Inequality

For any numbers a , b , and c, if a < b , then a < b , then

For any numbers a , b , and c, if a > b , then a > b , then

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.

What happens to an inequality when we divide or multiply both sides by a constant?

Let’s first multiply and divide both sides by a positive number.

The inequality signs stayed the same.

Does the inequality stay the same when we divide or multiply by a negative number?

Notice that when we filled in the inequality signs, the inequality signs reversed their direction.

When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.

This gives us the Multiplication and Division Property of Inequality.

## Multiplication and Division Property of Inequality

For any numbers a , b , and c ,

When we divide or multiply an inequality by a :

- positive number, the inequality stays the same.
- negative number, the inequality reverses.

Sometimes when solving an inequality, as in the next example, the variable ends upon the right. We can rewrite the inequality in reverse to get the variable to the left.

Think about it as “If Xander is taller than Andy, then Andy is shorter than Xander.”

## Example 2.50

Solve each inequality. Graph the solution on the number line, and write the solution in interval notation.

ⓐ x − 3 8 ≤ 3 4 x − 3 8 ≤ 3 4 ⓑ 9 y < 54 9 y < 54 ⓒ −15 < 3 5 z −15 < 3 5 z

## Try It 2.99

Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

ⓐ p − 3 4 ≥ 1 6 p − 3 4 ≥ 1 6 ⓑ 9 c > 72 9 c > 72 ⓒ 24 ≤ 3 8 m 24 ≤ 3 8 m

## Try It 2.100

ⓐ r − 1 3 ≤ 7 12 r − 1 3 ≤ 7 12 ⓑ 12 d ≤ 60 12 d ≤ 60 ⓒ −24 < 4 3 n −24 < 4 3 n

Be careful when you multiply or divide by a negative number—remember to reverse the inequality sign.

## Example 2.51

Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

ⓐ −13 m ≥ 65 −13 m ≥ 65 ⓑ n −2 ≥ 8 n −2 ≥ 8

## Try It 2.101

ⓐ −8 q < 32 −8 q < 32 ⓑ k −12 ≤ 15 . k −12 ≤ 15 .

## Try It 2.102

ⓐ −7 r ≤ − 70 −7 r ≤ − 70 ⓑ u −4 ≥ −16 . u −4 ≥ −16 .

Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but make sure to pay close attention when we multiply or divide to isolate the variable.

## Example 2.52

Solve the inequality 6 y ≤ 11 y + 17 , 6 y ≤ 11 y + 17 , graph the solution on the number line, and write the solution in interval notation.

## Try It 2.103

Solve the inequality, graph the solution on the number line, and write the solution in interval notation: 3 q ≥ 7 q − 23 . 3 q ≥ 7 q − 23 .

## Try It 2.104

Solve the inequality, graph the solution on the number line, and write the solution in interval notation: 6 x < 10 x + 19 . 6 x < 10 x + 19 .

When solving inequalities, it is usually easiest to collect the variables on the side where the coefficient of the variable is largest. This eliminates negative coefficients and so we don’t have to multiply or divide by a negative—which means we don’t have to remember to reverse the inequality sign.

## Example 2.53

Solve the inequality 8 p + 3 ( p − 12 ) > 7 p − 28 , 8 p + 3 ( p − 12 ) > 7 p − 28 , graph the solution on the number line, and write the solution in interval notation.

## Try It 2.105

Solve the inequality 9 y + 2 ( y + 6 ) > 5 y − 24 9 y + 2 ( y + 6 ) > 5 y − 24 , graph the solution on the number line, and write the solution in interval notation.

## Try It 2.106

Solve the inequality 6 u + 8 ( u − 1 ) > 10 u + 32 6 u + 8 ( u − 1 ) > 10 u + 32 , graph the solution on the number line, and write the solution in interval notation.

Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.

## Example 2.54

Solve the inequality 8 x − 2 ( 5 − x ) < 4 ( x + 9 ) + 6 x , 8 x − 2 ( 5 − x ) < 4 ( x + 9 ) + 6 x , graph the solution on the number line, and write the solution in interval notation.

## Try It 2.107

Solve the inequality 4 b − 3 ( 3 − b ) > 5 ( b − 6 ) + 2 b 4 b − 3 ( 3 − b ) > 5 ( b − 6 ) + 2 b , graph the solution on the number line, and write the solution in interval notation.

## Try It 2.108

Solve the inequality 9 h − 7 ( 2 − h ) < 8 ( h + 11 ) + 8 h 9 h − 7 ( 2 − h ) < 8 ( h + 11 ) + 8 h , graph the solution on the number line, and write the solution in interval notation.

We can clear fractions in inequalities much as we did in equations. Again, be careful with the signs when multiplying or dividing by a negative.

## Example 2.55

Solve the inequality 1 3 a − 1 8 a > 5 24 a + 3 4 , 1 3 a − 1 8 a > 5 24 a + 3 4 , graph the solution on the number line, and write the solution in interval notation.

## Try It 2.109

Solve the inequality 1 4 x − 1 12 x > 1 6 x + 7 8 1 4 x − 1 12 x > 1 6 x + 7 8 , graph the solution on the number line, and write the solution in interval notation.

## Try It 2.110

Solve the inequality 2 5 z − 1 3 z < 1 15 z − 3 5 2 5 z − 1 3 z < 1 15 z − 3 5 , graph the solution on the number line, and write the solution in interval notation.

Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like “more than” and “less than.” But others are not as obvious. Table 2.2 shows some common phrases that indicate inequalities.

## Example 2.56

Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.

## Try It 2.111

Nineteen less than p is no less than 47.

## Try It 2.112

Four more than a is at most 15.

Solve Applications with Linear Inequalities

Many real-life situations require us to solve inequalities. The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations.

We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.

Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down.

## Example 2.57

Dawn won a mini-grant of $4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost $254.12 each, including tax and delivery. What is the maximum number of tablets Dawn can buy?

## Try It 2.113

Angie has $20 to spend on juice boxes for her son’s preschool picnic. Each pack of juice boxes costs $2.63. What is the maximum number of packs she can buy?

## Try It 2.114

Daniel wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost $42.75 per person for dinner, including tip and tax. His budget for the party is $500. What is the maximum number of people Daniel can have at the party?

## Example 2.58

Taleisha’s phone plan costs her $28.80 a month plus $0.20 per text message. How many text messages can she send/receive and keep her monthly phone bill no more than $50?

## Try It 2.115

Sergio and Lizeth have a very tight vacation budget. They plan to rent a car from a company that charges $75 a week plus $0.25 a mile. How many miles can they travel during the week and still keep within their $200 budget?

## Try It 2.116

Rameen’s heating bill is $5.42 per month plus $1.08 per therm. How many therms can Rameen use if he wants his heating bill to be a maximum of $87.50.

Profit is the money that remains when the costs have been subtracted from the revenue. In the next example, we will find the number of jobs a small businesswoman needs to do every month in order to make a certain amount of profit.

## Example 2.59

Felicity has a calligraphy business. She charges $2.50 per wedding invitation. Her monthly expenses are $650. How many invitations must she write to earn a profit of at least $2,800 per month?

## Try It 2.117

Caleb has a pet sitting business. He charges $32 per hour. His monthly expenses are $2,272. How many hours must he work in order to earn a profit of at least $800 per month?

## Try It 2.118

Elliot has a landscape maintenance business. His monthly expenses are $1,100. If he charges $60 per job, how many jobs must he do to earn a profit of at least $4,000 a month?

There are many situations in which several quantities contribute to the total expense. We must make sure to account for all the individual expenses when we solve problems like this.

## Example 2.60

Malik is planning a six-day summer vacation trip. He has $840 in savings, and he earns $45 per hour for tutoring. The trip will cost him $525 for airfare, $780 for food and sightseeing, and $95 per night for the hotel. How many hours must he tutor to have enough money to pay for the trip?

## Try It 2.119

Brenda’s best friend is having a destination wedding and the event will last three days. Brenda has $500 in savings and can earn $15 an hour babysitting. She expects to pay $350 airfare, $375 for food and entertainment and $60 a night for her share of a hotel room. How many hours must she babysit to have enough money to pay for the trip?

## Try It 2.120

Josue wants to go on a 10-night road trip with friends next spring. It will cost him $180 for gas, $450 for food, and $49 per night to share a motel room. He has $520 in savings and can earn $30 per driveway shoveling snow. How many driveways must he shovel to have enough money to pay for the trip?

## Section 2.5 Exercises

Practice makes perfect.

In the following exercises, graph each inequality on the number line and write in interval notation.

ⓐ x < −2 x < −2 ⓑ x ≥ −3.5 x ≥ −3.5 ⓒ x ≤ 2 3 x ≤ 2 3

ⓐ x > 3 x > 3 ⓑ x ≤ −0.5 x ≤ −0.5 ⓒ x ≥ 1 3 x ≥ 1 3

ⓐ x ≥ −4 x ≥ −4 ⓑ x < 2.5 x < 2.5 ⓒ x > − 3 2 x > − 3 2

ⓐ x ≤ 5 x ≤ 5 ⓑ x ≥ −1.5 x ≥ −1.5 ⓒ x < − 7 3 x < − 7 3

ⓐ −5 < x < 2 −5 < x < 2 ⓑ −3 ≤ x < 1 −3 ≤ x < 1 ⓒ 0 ≤ x ≤ 1.5 0 ≤ x ≤ 1.5

ⓐ −2 < x < 0 −2 < x < 0 ⓑ −5 ≤ x < −3 −5 ≤ x < −3 ⓒ 0 ≤ x ≤ 3.5 0 ≤ x ≤ 3.5

ⓐ −1 < x < 3 −1 < x < 3 ⓑ −3 < x ≤ −2 −3 < x ≤ −2 ⓒ −1.25 ≤ x ≤ 0 −1.25 ≤ x ≤ 0

ⓐ −4 < x < 2 −4 < x < 2 ⓑ −5 < x ≤ −2 −5 < x ≤ −2 ⓒ −3.75 ≤ x ≤ 0 −3.75 ≤ x ≤ 0

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

ⓐ a + 3 4 ≥ 7 10 a + 3 4 ≥ 7 10 ⓑ 8 x > 72 8 x > 72 ⓒ 20 > 2 5 h 20 > 2 5 h

ⓐ b + 7 8 ≥ 1 6 b + 7 8 ≥ 1 6 ⓑ 6 y < 48 6 y < 48 ⓒ 40 < 5 8 k 40 < 5 8 k

ⓐ f − 13 20 < − 5 12 f − 13 20 < − 5 12 ⓑ 9 t ≥ −27 9 t ≥ −27 ⓒ 7 6 j ≥ 42 7 6 j ≥ 42

ⓐ g − 11 12 < − 5 18 g − 11 12 < − 5 18 ⓑ 7 s < −28 7 s < −28 ⓒ 9 4 g ≤ 36 9 4 g ≤ 36

ⓐ −5 u ≥ 65 −5 u ≥ 65 ⓑ a −3 ≤ 9 a −3 ≤ 9

ⓐ −8 v ≤ 96 −8 v ≤ 96 ⓑ b −10 ≥ 30 b −10 ≥ 30

ⓐ −9 c < 126 −9 c < 126 ⓑ −25 < p −5 −25 < p −5

ⓐ −7 d > 105 −7 d > 105 ⓑ −18 > q −6 −18 > q −6

4 v ≥ 9 v − 40 4 v ≥ 9 v − 40

5 u ≤ 8 u − 21 5 u ≤ 8 u − 21

13 q < 7 q − 29 13 q < 7 q − 29

9 p > 14 p − 18 9 p > 14 p − 18

12 x + 3 ( x + 7 ) > 10 x − 24 12 x + 3 ( x + 7 ) > 10 x − 24

9 y + 5 ( y + 3 ) < 4 y − 35 9 y + 5 ( y + 3 ) < 4 y − 35

6 h − 4 ( h − 1 ) ≤ 7 h − 11 6 h − 4 ( h − 1 ) ≤ 7 h − 11

4 k − ( k − 2 ) ≥ 7 k − 26 4 k − ( k − 2 ) ≥ 7 k − 26

8 m − 2 ( 14 − m ) ≥ 7 ( m − 4 ) + 3 m 8 m − 2 ( 14 − m ) ≥ 7 ( m − 4 ) + 3 m

6 n − 12 ( 3 − n ) ≤ 9 ( n − 4 ) + 9 n 6 n − 12 ( 3 − n ) ≤ 9 ( n − 4 ) + 9 n

3 4 b − 1 3 b < 5 12 b − 1 2 3 4 b − 1 3 b < 5 12 b − 1 2

9 u + 5 ( 2 u − 5 ) ≥ 12 ( u − 1 ) + 7 u 9 u + 5 ( 2 u − 5 ) ≥ 12 ( u − 1 ) + 7 u

2 3 g − 1 2 ( g − 14 ) ≤ 1 6 ( g + 42 ) 2 3 g − 1 2 ( g − 14 ) ≤ 1 6 ( g + 42 )

4 5 h − 2 3 ( h − 9 ) ≥ 1 15 ( 2 h + 90 ) 4 5 h − 2 3 ( h − 9 ) ≥ 1 15 ( 2 h + 90 )

5 6 a − 1 4 a > 7 12 a + 2 3 5 6 a − 1 4 a > 7 12 a + 2 3

12 v + 3 ( 4 v − 1 ) ≤ 19 ( v − 2 ) + 5 v 12 v + 3 ( 4 v − 1 ) ≤ 19 ( v − 2 ) + 5 v

15 k ≤ −40 15 k ≤ −40

35 k ≥ −77 35 k ≥ −77

23 p − 2 ( 6 − 5 p ) > 3 ( 11 p − 4 ) 23 p − 2 ( 6 − 5 p ) > 3 ( 11 p − 4 )

18 q − 4 ( 10 − 3 q ) < 5 ( 6 q − 8 ) 18 q − 4 ( 10 − 3 q ) < 5 ( 6 q − 8 )

− 9 4 x ≥ − 5 12 − 9 4 x ≥ − 5 12

− 21 8 y ≤ − 15 28 − 21 8 y ≤ − 15 28

c + 34 < −99 c + 34 < −99

d + 29 > −61 d + 29 > −61

m 18 ≥ −4 m 18 ≥ −4

n 13 ≤ −6 n 13 ≤ −6

In the following exercises, translate and solve. Then graph the solution on the number line and write the solution in interval notation.

Three more than h is no less than 25.

Six more than k exceeds 25.

Ten less than w is at least 39.

Twelve less than x is no less than 21.

Negative five times r is no more than 95.

Negative two times s is lower than 56.

Nineteen less than b is at most −22 . −22 .

Fifteen less than a is at least −7 . −7 .

In the following exercises, solve.

Alan is loading a pallet with boxes that each weighs 45 pounds. The pallet can safely support no more than 900 pounds. How many boxes can he safely load onto the pallet?

The elevator in Yehire’s apartment building has a sign that says the maximum weight is 2100 pounds. If the average weight of one person is 150 pounds, how many people can safely ride the elevator?

Andre is looking at apartments with three of his friends. They want the monthly rent to be no more than $2,360. If the roommates split the rent evenly among the four of them, what is the maximum rent each will pay?

Arleen got a $20 gift card for the coffee shop. Her favorite iced drink costs $3.79. What is the maximum number of drinks she can buy with the gift card?

Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him $10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?

Ryan charges his neighbors $17.50 to wash their car. How many cars must he wash next summer if his goal is to earn at least $1,500?

Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?

Kimuyen needs to earn $4,150 per month in order to pay all her expenses. Her job pays her $3,475 per month plus 4% of her total sales. What is the minimum Kimuyen’s total sales must be in order for her to pay all her expenses?

Andre has been offered an entry-level job. The company offered him $48,000 per year plus 3.5% of his total sales. Andre knows that the average pay for this job is $62,000. What would Andre’s total sales need to be for his pay to be at least as high as the average pay for this job?

Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?

Jake’s water bill is $24.80 per month plus $2.20 per ccf (hundred cubic feet) of water. What is the maximum number of ccf Jake can use if he wants his bill to be no more than $60?

Kiyoshi’s phone plan costs $17.50 per month plus $0.15 per text message. What is the maximum number of text messages Kiyoshi can use so the phone bill is no more than $56.60?

Marlon’s TV plan costs $49.99 per month plus $5.49 per first-run movie. How many first-run movies can he watch if he wants to keep his monthly bill to be a maximum of $100?

Kellen wants to rent a banquet room in a restaurant for her cousin’s baby shower. The restaurant charges $350 for the banquet room plus $32.50 per person for lunch. How many people can Kellen have at the shower if she wants the maximum cost to be $1,500?

Moshde runs a hairstyling business from her house. She charges $45 for a haircut and style. Her monthly expenses are $960. She wants to be able to put at least $1,200 per month into her savings account order to open her own salon. How many “cut & styles” must she do to save at least $1,200 per month?

Noe installs and configures software on home computers. He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?

Katherine is a personal chef. She charges $115 per four-person meal. Her monthly expenses are $3,150. How many four-person meals must she sell in order to make a profit of at least $1,900?

Melissa makes necklaces and sells them online. She charges $88 per necklace. Her monthly expenses are $3,745. How many necklaces must she sell if she wants to make a profit of at least $1,650?

Five student government officers want to go to the state convention. It will cost them $110 for registration, $375 for transportation and food, and $42 per person for the hotel. There is $450 budgeted for the convention in the student government savings account. They can earn the rest of the money they need by having a car wash. If they charge $5 per car, how many cars must they wash in order to have enough money to pay for the trip?

Cesar is planning a four-day trip to visit his friend at a college in another state. It will cost him $198 for airfare, $56 for local transportation, and $45 per day for food. He has $189 in savings and can earn $35 for each lawn he mows. How many lawns must he mow to have enough money to pay for the trip?

Alonzo works as a car detailer. He charges $175 per car. He is planning to move out of his parents’ house and rent his first apartment. He will need to pay $120 for application fees, $950 for security deposit, and first and last months’ rent at $1,140 per month. He has $1,810 in savings. How many cars must he detail to have enough money to rent the apartment?

Eun-Kyung works as a tutor and earns $60 per hour. She has $792 in savings. She is planning an anniversary party for her parents. She would like to invite 40 guests. The party will cost her $1,520 for food and drinks and $150 for the photographer. She will also have a favor for each of the guests, and each favor will cost $7.50. How many hours must she tutor to have enough money for the party?

## Everyday Math

Maximum load on a stage In 2014, a high school stage collapsed in Fullerton, California, when 250 students got on stage for the finale of a musical production. Two dozen students were injured. The stage could support a maximum of 12,750 pounds. If the average weight of a student is assumed to be 140 pounds, what is the maximum number of students who could safely be on the stage?

Maximum weight on a boat In 2004, a water taxi sank in Baltimore harbor and five people drowned. The water taxi had a maximum capacity of 3,500 pounds (25 people with average weight 140 pounds). The average weight of the 25 people on the water taxi when it sank was 168 pounds per person. What should the maximum number of people of this weight have been?

Wedding budget Adele and Walter found the perfect venue for their wedding reception. The cost is $9850 for up to 100 guests, plus $38 for each additional guest. How many guests can attend if Adele and Walter want the total cost to be no more than $12,500?

Shower budget Penny is planning a baby shower for her daughter-in-law. The restaurant charges $950 for up to 25 guests, plus $31.95 for each additional guest. How many guests can attend if Penny wants the total cost to be no more than $1,500?

## Writing Exercises

Explain why it is necessary to reverse the inequality when solving −5 x > 10 . −5 x > 10 .

Explain why it is necessary to reverse the inequality when solving n −3 < 12 . n −3 < 12 .

Find your last month’s phone bill and the hourly salary you are paid at your job. Calculate the number of hours of work it would take you to earn at least enough money to pay your phone bill by writing an appropriate inequality and then solving it. Do you feel this is an appropriate number of hours? Is this the appropriate phone plan for you?

Find out how many units you have left, after this term, to achieve your college goal and estimate the number of units you can take each term in college. Calculate the number of terms it will take you to achieve your college goal by writing an appropriate inequality and then solving it. Is this an acceptable number of terms until you meet your goal? What are some ways you could accelerate this process?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction

- Authors: Lynn Marecek, Andrea Honeycutt Mathis
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- Book title: Intermediate Algebra 2e
- Publication date: May 6, 2020
- Location: Houston, Texas
- Book URL: https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
- Section URL: https://openstax.org/books/intermediate-algebra-2e/pages/2-5-solve-linear-inequalities

© Jul 7, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

- Solve equations and inequalities
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## What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

- The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
- The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
- The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
- The calculus section will carry out differentiation as well as definite and indefinite integration.
- The matrices section contains commands for the arithmetic manipulation of matrices.
- The graphs section contains commands for plotting equations and inequalities.
- The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

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## Course: 6th grade > Unit 7

- Testing solutions to inequalities
- Testing solutions to inequalities (basic)
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## Inequalities word problems

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## Corbettmaths

Videos, worksheets, 5-a-day and much more, inequalities practice questions, click here for questions, click here for answers.

## Math Worksheets

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## Looking for Something?

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## Inequalities Worksheets

Are you looking for free math worksheets that will help your students develop and master real-life math skills? The algebra worksheets below will introduce your students to solving inequalities and graphing inequalities. As they take a step-by-step approach to solving inequalities, they will also practice other essential algebra skills like using inverse operations to solve equations.

Solving Inequalities Worksheet 1 – Here is a twelve problem worksheet featuring simple one-step inequalities. Use inverse operations or mental math to solve for x . Solving Inequalities Worksheet 1 RTF Solving Inequalities Worksheet 1 PDF Preview Solving Inequalities Worksheet 1 in Your Browser View Answers

Solving Inequalities Worksheet 2 – Here is a twelve problem worksheet featuring simple one-step inequalities. Use inverse operations or mental math to solve for x . Solving Inequalities Worksheet 2 RTF Solving Inequalities Worksheet 2 PDF Preview Solving Inequalities Worksheet 2 in Your Browser View Answers

Solving Inequalities Worksheet 3 – Here is a twelve problem worksheet featuring two-step inequalities. Use inverse operations or mental math to solve for x . Solving Inequalities Worksheet 3 RTF Solving Inequalities Worksheet 3 PDF Preview Solving Inequalities Worksheet 3 in Your Browser View Answers

Solving Inequalities Worksheet 4 – Here is a twelve problem worksheet featuring one-step inequalities. Use inverse operations or mental math to solve for x . Solving Inequalities Worksheet 4 RTF Solving Inequalities Worksheet 4 PDF Preview Solving Inequalities Worksheet 4 in Your Browser View Answers

Solving Inequalities Worksheet 5 – Here is a twelve problem worksheet featuring two-step inequalities. Use inverse operations or mental math to solve for x . Solving Inequalities Worksheet 5 RTF Solving Inequalities Worksheet 5 PDF Preview Solving Inequalities Worksheet 5 in Your Browser View Answers

Graphing Inequalities Workheet 1 – Here is a 15 problem worksheet where students will graph simple inequalities like “ x < -2″ on a number line. Graphing Inequalities Worksheet 1 RTF Graphing Inequalities 1 PDF View Answers

Graphing Inequalities Workheet 2 – Here is a 15 problem worksheet where students will graph simple inequalities like “ x < -2″ and “ -x > 2″ on a number line. Be careful, you may have to reverse one or two of the inequality symbols to get the correct solution set. Graphing Inequalities 2 RTF Graphing Inequalities 2 PDF View Answers

Graphing Inequalities Workheet 3 – Here is a 12 problem worksheet where students will both solve inequalities and graph inequalities on a number line. This set features one-step addition inequalities such as “x + 5 > 7”. Graphing Inequalities 3 RTF Graphing Inequalities 3 PDF View Answers

Graphing Inequalities Workheet 4 – Here is a 12 problem worksheet where students will both solve inequalities and graph inequalities on a number line. This set features one-step addition and subtraction inequalities such as “5 + x > 7” and “x – 3″ < 21”. Graphing Inequalities 4 RTF Graphing Inequalities 4 PDF View Answers

Graphing Inequalities Workheet 5 – Here is a 12 problem worksheet where students will both solve inequalities and graph inequalities on a number line. This set features two-step addition inequalities such as “2x + 5 > 15” . Graphing Inequalities 5 RTF Graphing Inequalities 5 PDF View Answers

Graphing Inequalities Workheet 6 – Here is a 12 problem worksheet where students will both solve inequalities and graph inequalities on a number line. This set features two-step addition and subtraction inequalities such as “2x + 5 > 15” and “ 4x -2 = 14. Graphing Inequalities 6 RTF Graphing Inequalities 6 PDF View Answers

## Absolute Value Inequality Worksheets (Single Variable)

Absolute Value Inequality Worksheet 1 – Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities. These are one-step inequalities with mostly positive integers. Absolute Value Equations Worksheet 1 RTF Absolute Value Equations Worksheet 1 PDF View Answers

Absolute Value Inequality Worksheet 2 – Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities. These are one-step inequalities where you’ll need to use all of your inverse operations knowledge. Absolute Value Equations Worksheet 2 RTF Absolute Value Equations Worksheet 2 PDF View Answers

Absolute Value Inequality Worksheet 3 – Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities. These are two-step inequalities where you’ll need to use all of your inverse operations knowledge. Absolute Value Equations Worksheet 3 RTF Absolute Value Equations Worksheet 3 PDF View Answers

Absolute Value Inequality Worksheet 4 – Here is a 9 problem worksheet where you will find the solution set of absolute value inequalities. These are two-step inequalities that can get quite complicated. A nice challenge for your higher-level learners. Absolute Value Equations Worksheet 4 RTF Absolute Value Equations Worksheet 4 PDF View Answers

## 20 Comments

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## Mr. Colwell

I’m glad that I could help. Thanks for visiting the site and come back often!

## Kaity Fanara

Very helpful, thanks!

You’re very welcome. I plan to add some Graphing Inequalities Worksheets soon. Thanks for your support.

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That’s a pretty good one. Let me see what I can do.

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It’s nice to hear from our international visitors. I’m always adding new content. Check back often!

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Home / United States / Math Classes / Worksheets / Solving Inequalities Worksheets

## Solving Inequalities Worksheets

Inequalities in mathematics demonstrate a relationship between two values when these two values are unequal. These value s can be numbers or algebraic expressions. When practicing with solving inequalities worksheets, students are presented with an engaging and fun way to learn inequalities and solve problems at their own pace. Solving inequalities worksheets can guide students through graphical representations using real-life examples and relatable situations. ...Read More Read Less

## The Benefits of Solving Inequalities Worksheets:

Analyze boundaries :

For situations where exact equality is not achieved, inequalities can help students assess boundary values of the required result. Even though students do not get the precise result, they can identify upper or lower boundaries.

Flexibility in prediction :

This leaves room for flexibility for scenarios where changes might occur, therefore lowering the possibility of error that may have been generated from the prediction of a fixed outcome.

Real world application :

Solving inequalities worksheets can show students how these boundaries and predictions can be implemented in our daily lives through mathematical solving of linear inequalities.

- Interactive Worksheets
- Printable Worksheets

## 7th Grade Solving Inequalities Worksheets

- 7th Grade Mathematical Structure of Word Problems Worksheets

## Choose Math Worksheets by Grade

Choose math worksheets by topic, solving inequalities worksheets explained:.

While solving simple inequalities or linear inequalities in mathematics, there are four possible results for the comparison of the left-hand side (LHS) to the right-hand side (RHS) of an inequality sign:

- Less than (<): LHS is always lesser in value than RHS
- Less than or equal to ( ? ): LHS is lesser or equal in value to RHS
- Greater than (>): LHS is always greater in value than RHS
- Greater than or equal to ( ? ): LHS is greater or equal in value to RHS

Students are provided with real-life examples and can learn to solve inequalities through these worksheets.

Ravi was asked to buy a maximum of 80 chocolates. These chocolates are sold in packets, each of which contains 12 chocolates. Find the number of packets he needs to buy to satisfy the requirement.

Assume that Ravi has to buy x number of packets to meet the requirements. Each packet contains 12 chocolates.

Thus we can write the above problem in the form of linear inequality as:

12 * number of packets < 80

12x < 80

=> 12x/12 < 80/12 (Dividing both sides with 12)

=> x < 20/3

Therefore the number of packets Ravi needs to buy to meet the requirement is 20/3.

Or, x < 6.66 (as 20/3 is equal to 6.66)

Because the number of packets cannot be in a fraction, we must round it up to 6 packets.

This is how a scenario can be represented in the form of inequalities.

Inequalities can be solved by either of the following methods:

- Addition or subtraction of the same number from both sides of the inequality sign without altering the sign
- Multiplication or division of the same non-zero positive number to both sides of the sign. In the case of multiplication or division with a negative number, the inequality sign will be reversed.

## Printable Solving inequalities Worksheets

Solving inequalities worksheets can be downloaded or printed so students can practice with them at their own pace and convenience. The printable worksheets contain varying levels of difficulty, from simple inequalities to more complex ones, to assist students in understanding this concept.

## What are the solving inequalities worksheets about?

Solving inequalities worksheets are free printable fun and interactive worksheets that contain questions based on inequalities. The worksheets have real-life questions that will help students in connecting inequality with real-life situations.

## What is inequality?

Inequality is the representation of two algebraic expressions which are not equal. We use “<” or “>” signs to represent inequality. Students can find good practice questions in the solving inequalities worksheets.

## How is inequality differ from equations?

The equation is used to represent two algebraic expressions when the are equal but inequality is used to represent two algebraic expressions when the are not equal. For equations we use “=” sign but for inequality we use “<” or “>”.

## What are the methods used in solving inequalities worksheets to solve questions?

The methods used in solving inequalities worksheets for solutions are the graphical method, using addition, subtraction, multiplication, and division method, using the two-step method. By using all these methods a stepwise solution is provided for a better understanding of the concept.

## How are the solving inequalities worksheets beneficial for students?

Solving inequalities worksheets have different kinds of questions based on inequality that will improve the accuracy level of the student. It will enhance the knowledge of inequalities and their applications of them. We have stepwise solutions for questions that will improve students’ self-assessment skills and solution writing and reinforce their understanding of the concept. Students will get to know how to solve the same kind of question with different methods. Worksheets have images that will improve the visualization of concepts.

## Solving Inequalities Worksheets

Inequalities come in as a new territory to explore since so far we have always tried to get an answer. In contrary to the general notion of getting one answer, solving inequalities might end up at getting one, more, or no answers at all. Multiple answers mostly come as a range but it should not come as a surprise as all those values hold true for the given inequality.

## Benefits of Worksheet on Solving Inequalities

When you are doing worksheets on solving inequalities, you become fluent with the process. In order to solve an inequality, you need to perform a certain number of arithmetic operations and in the beginning, it might appear as cumbersome. Solving a few questions would make it easier for you to face and solve the question when it actually comes in front of you. It will help you get an edge over the rest of the students.

Read More :- Topic-wise Math Worksheets

## Download Solving Inequalities Worksheet PDFs

## KAIST Math Problem of the Week

Weekly math challenges in kaist.

## Solution: 2023-15 An inequality for complex polynomials

Let \(p(z), q(z) \) and \(r(z)\) be polynomials with complex coefficients in the complex plane. Suppose that \(|p(z)| + |q(z)| \leq |r(z)|\) for every \(z\). Show that there exist two complex numbers \( a,b \) such that \(|a|^2 +|b|^2 =1\) and \( a p(z) + bq(z) =0 \) for every \(z\).

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Here is the best solution of problem 2023-15 .

Other solutions were submitted by 강지민 (세마고 3학년, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 신민서 (KAIST 수리과학과 20학번, +3), 여인영 (KAIST 물리학과 20학번, +3),이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 조현준 (KAIST 수리과학과 22학번, +4), 지은성 (KAIST 수리과학과 20학번, +3), 최민규 (한양대학교 의과대학 졸업생, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), Muhammadfiruz Hasanov (+3).

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## Press release

Ey engages young talent to develop ai solutions to solve real-world problems, new delhi, 27 september 2023: ey, the leading professional services organization, launched the fourth edition of its flagship college campus challenge techathon4.0. focused on the theme of generative ai (genai), the competition aims to address real-world challenges focused on revolutionizing, transforming, and enhancing the working world using the power of ai technologies..

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## Show resources

Ivca ey monthly pe vc roundup august 2023.

- Launches the fourth edition of its annual flagship challenge ‘Techathon’.
- Aims to expand its talent pool for its growing Technology Consulting practice; 1,300+ graduates hired from campuses in last six months alone.

New Delhi, 27 September 2023 : EY, the leading professional services organization, launched the fourth edition of its flagship college campus challenge Techathon4.0 . Focused on the theme of Generative AI (GenAI), the competition aims to address real-world challenges focused on revolutionizing, transforming, and enhancing the working world using the power of AI technologies.

Students will be provided access to EY proprietary technologies to help them develop GenAI-based solutions to solve problems across Healthcare, Banking, Financial Services and Insurance (BFSI), Education, Retail, and Media & Entertainment sectors.

Mahesh Makhija, Technology Consulting Leader, EY India , said, “As an organization deeply committed to pioneering tech-driven innovations and solutions, we are excited about the transformative potential of Gen AI. Organizations have already started building use cases with GenAI but will require focused effort on moving from a single use case to organization-wide deployments to create value.

With the adoption of emerging technologies, it has become even more imperative to focus on building quality talent. Techathon is not just another competition; it’s a reflection of our commitment to hone talent within EY and in the larger ecosystem.”

EY India’s tech talent has tripled at a 44% CAGR over the last three years. The past (calendar) year alone has witnessed a remarkable surge in campus graduate intake, exemplifying EY’s commitment to nurturing fresh talent from over 60 premier colleges across various business verticals.

About Techathon4.0

The EY Techathon challenge began in 2020 as a response to the urgent need for innovative solutions to vaccinate over a billion population to fight the COVID-19 pandemic. Later in 2021, the Challenge pivoted to technology-led sustainability solutions, encouraging eco-conscious innovation as the pandemic brought to the fore the climate change challenges facing the planet. The third edition of EY Techathon in 2022 challenged students across the country to develop industry-relevant solutions in the dynamic space of Metaverse.

This year’s Techathon is focused on Generative AI , a disruptive technology which is changing the future of work and gaining significant adoption among businesses. Aspiring participants are invited to register along with the team details and problem statement(s) they aspire to solve. The winners will be awarded cash prizes, mentorship by industry and EY leaders, and an opportunity to intern with EY India.

Past years' winners and runners-up have come from PES University, SRM Institute of Science and Technology, Shailesh J Mehta School of Management, KJ Somaiya College of Engineering, Thadomal Shahani Engineering College, and Vellore Institute of Technology.

Over the years Techathon has seen an eminent jury comprising industry experts and technology enthusiasts like R Chandrasekhar , Former Telecom and IT Secretary, Government of India, and Former President of NASSCOM, Dr. Rohini Srivathsa , CTO and National Technology Officer, Microsoft, Dr. Krishna Ella , Chairman and Managing Director, Bharat Biotech, Dr. Harish Iyer, Head of Digital and Health Innovation, Bill and Melinda Gates Foundation, Dr. Manish Pant , Chief, Health and Development, United Nations Development Programme, Aashish Kshetry , Vice President, Information Technology, Asian Paints; Bhuwan Lodha , Senior Vice President, Head of Digital, Mahindra & Mahindra; Deepak Bhosale , General Manager – Information Technology, Asian Paints; Madhavi Isanaka , SVP, Chief Digital Officer, Adani Green Energy & Adani New Industries, and, Manish Grover , Executive Director - Strategic Information Systems, IOCL, Bose Varghese , Head Green Initiatives, Infosys, Binu Treesa , IT Sustainability Leader India, Decathlon and Meenakshi Burra , CDO and VP Technology, Analytics & Business Services (South Asia), Hindustan Unilever.

To know more about Techathon 4.0: visit: Techathon 4.0 | Re-imagining possibilities with GenAI

Note to Editors

EY exists to build a better working world, helping create long-term value for clients, people and society and build trust in the capital markets. Enabled by data and technology, diverse EY teams in over 150 countries provide trust through assurance and help clients grow, transform and operate. Working across assurance, consulting, law, strategy, tax and transactions, EY teams ask better questions to find new answers for the complex issues facing our world today. EY refers to the global organization, and may refer to one or more, of the member firms of Ernst & Young Global Limited, each of which is a separate legal entity. Ernst & Young Global Limited, a UK company limited by guarantee, does not provide services to clients. Information about how EY collects and uses personal data and a description of the rights individuals have under data protection legislation are available via ey.com/privacy. EY member firms do not practice law where prohibited by local laws.

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## IMAGES

## VIDEO

## COMMENTS

How to Solve Solving inequalities is very like solving equations, we do most of the same things ... ... but we must also pay attention to the direction of the inequality. Direction: Which way the arrow "points" Some things can change the direction! < becomes > > becomes < ≤ becomes ≥ ≥ becomes ≤ Safe Things To Do

Basic inequalities do not need to be solved since the variable is already by itself. For example, consider the inequalities x>7 and y ≤-3. These inequalities can be though of as solved because the variables are already on their own. These "solved" inequalities are illustrated in Figure 02 below. Figure 02: How to solve inequalities.

There are lots of strategies we can use to solve equations. Let's explore some different ways to solve equations and inequalities. We'll also see what it takes for an equation to have no solution, or infinite solutions. Linear equations with variables on both sides Learn Why we do the same thing to both sides: Variable on both sides

Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality. Exercise 3.6.1 3.6. 1. Emma got a new job and will have to move. Her monthly income will be $5,265.

Type a math problem Solve Examples 3x + 4 > 6 x + y < 0 5 > 2x + 3 −2 < 3x + 2 < 8 2x2 ≥ 50 3x + 35 ≤ 8 3x+4 > 6 5 > 2x+3 2x2 ≥ 50 Learn about inequalities using our free math solver with step-by-step solutions.

How do you solve inequalities? To solve inequalities, isolate the variable on one side of the inequality, If you multiply or divide both sides by a negative number, flip the direction of the inequality. Show more Related Symbolab blog posts High School Math Solutions - Inequalities Calculator, Logarithmic Inequalities

To solve your inequality using the Inequality Calculator, type in your inequality like x+7>9. The inequality solver will then show you the steps to help you learn how to solve it on your own. Less Than Or Equal To Type <= for "less than or equal to". Here is an example: Greater Than Or Equal To Type >= for "greater than or equal to".

Algebra Inequality Calculator Step 1: Enter the inequality below which you want to simplify. The inequality calculator simplifies the given inequality. You will get the final answer in inequality form and interval notation. Step 2: Click the blue arrow to submit.

Example 2.48. Graph each inequality on the number line and write in interval notation. ⓐ x ≥ −3 ⓑ x < 2.5 ⓒ x ≤ − 3 5. Shade to the right of −3, and put a bracket at −3. Write in interval notation. Shade to the left of 2.5 and put a parenthesis at 2.5. Write in interval notation.

Solution Dividing each side by -3, we obtain Always check in the original equation. Another way of solving the equation 3x - 4 = 7x + 8 would be to first subtract 3x from both sides obtaining -4 = 4x + 8, then subtract 8 from both sides and get -12 = 4x. Now divide both sides by 4 obtaining - 3 = x or x = - 3. First remove parentheses.

We will also introduce interval notation. Polynomial Inequalities - In this section we will continue solving inequalities. However, in this section we move away from linear inequalities and move on to solving inequalities that involve polynomials of degree at least 2.

Step 3. Write the solution in interval notation. [ − 3, 2) All the numbers that make both inequalities true are the solution to the compound inequality. Example 2.7.2. Solve the compound inequality. Graph the solution and write the solution in interval notation: 4x − 7 < 9 and 5x + 8 ≥ 3.

QuickMath allows students to get instant solutions to all kinds of math problems, from algebra and equation solving right through to calculus and matrices. QuickMath Solve equations and ... The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.

Try: solve a linear inequality word problem. Zoey has \$5 $5 and wants to rent a scooter. The scooter costs \$1 $1 to unlock and \$0.25 $0.25 for each minute of use. Write an expression for the total cost in dollars of renting the scooter if Zoey uses it for x x minutes. \text {total cost}=\, total cost =. Since Zoey has \$5 $5, the cost of ...

Chuck Towle. The equation y>5 is a linear inequality equation. y=0x + 5. So whatever we put in for x, we get x*0 which always = 0. So for whatever x we use, y always equals 5. The same thing is true for y>5. y > 0x + 5. And again, no matter what x we use, y is always greater than 5.

For problems 1 - 6 solve each of the following inequalities. Give the solution in both inequality and interval notations. 4(z +2)−1 > 5 −7(4 −z) 4 ( z + 2) − 1 > 5 − 7 ( 4 − z) Solution 1 2(3 +4t) ≤ 6(1 3 − 1 2 t)− 1 4 (2+10t) 1 2 ( 3 + 4 t) ≤ 6 ( 1 3 − 1 2 t) − 1 4 ( 2 + 10 t) Solution −1 <4x+2 <10 − 1 < 4 x + 2 < 10 Solution

Algebra 1 Course: Algebra 1 > Unit 2 Lesson 5: Multi-step inequalities Inequalities with variables on both sides Inequalities with variables on both sides (with parentheses) Multi-step inequalities Multi-step linear inequalities Using inequalities to solve problems Using inequalities to solve problems Math > Algebra 1 >

6th grade Course: 6th grade > Unit 7 Lesson 7: Intro to inequalities with variables Testing solutions to inequalities Testing solutions to inequalities (basic) Plotting inequalities Plotting an inequality example Graphing basic inequalities Inequality from graph Plotting inequalities Inequalities word problems Inequalities word problems

The Corbettmaths Practice Questions on inequalities. Videos, worksheets, 5-a-day and much more

View Answers Solving Inequalities Worksheet 4 - Here is a twelve problem worksheet featuring one-step inequalities. Use inverse operations or mental math to solve for x. Solving Inequalities Worksheet 4 RTF Solving Inequalities Worksheet 4 PDF Preview Solving Inequalities Worksheet 4 in Your Browser View Answers

When practicing with solving inequalities worksheets, students are presented with an engaging and fun way to learn inequalities and solve problems at their own pace. ... The methods used in solving inequalities worksheets for solutions are the graphical method, using addition, subtraction, multiplication, and division method, using the two-step ...

Multiple answers mostly come as a range but it should not come as a surprise as all those values hold true for the given inequality. Benefits of Worksheet on Solving Inequalities. When you are doing worksheets on solving inequalities, you become fluent with the process. In order to solve an inequality, you need to perform a certain number of ...

KAIST Math Problem of the Week Weekly Math Challenges in KAIST Menu Skip to ... Rules; Solution: 2023-15 An inequality for complex polynomials. Leave a reply. Let \(p(z), q(z) \) and \(r(z)\) be polynomials with complex coefficients in the complex plane. ... Here is the best solution of problem 2023-15. Other solutions were submitted by ...

The radical two-state solution would make it possible to direct resources to the reconstruction of society and the state, and also to repair Israel's status around the world. It would provide a proper response to antisemitism and help maintain the security of Diaspora Jews and the relationship with them. A return of the settlers would provide ...

Students will be provided access to EY proprietary technologies to help them develop GenAI-based solutions to solve problems across Healthcare, Banking, Financial Services and Insurance (BFSI), Education, Retail, and Media & Entertainment sectors. Mahesh Makhija, Technology Consulting Leader, EY India, said, "As an organization deeply ...