Inequalities Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Inequalities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Mathematical statements that compare two expressions using symbols like $<$ , $>$ , $\leq$ , or $\geq$ , indicating that one quantity is less than, greater than, or not equal to another. Unlike equations, inequalities describe a range of possible solutions.

Instead of 'equals exactly,' it's 'at least,' 'at most,' or 'greater/less than.'

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An inequality compares two expressions with $<,>,\le,\ge$ and describes a whole range of true values.

Common stuck point: The procedure for inequalities is the easy part; the trap is forgetting to flip the symbol when multiplying or dividing by a negative. Asking "Is the relation 'less/greater than (or equal)' so the answer is a range, not a single value?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the relation 'less/greater than (or equal)' so the answer is a range, not a single value?

Worked Examples

Example 1

easy

Solve

2x + 5 > 11

Solution set of 2x + 5 > 11 graphed on the number line

Answer

x > 3

First step

Subtract 5 from both sides:

2x > 6

Full solution

2
Divide both sides by 2: $x > 3$ .
3
The solution is all values greater than 3.

Solving inequalities follows the same steps as equations, with one key difference: multiplying or dividing by a negative number reverses the inequality sign.

Example 2

medium

Solve

-3x + 4 \leq 13

Solution set of −3x + 4 ≤ 13 graphed on the number line

Example 3

easy

Solve

-4x > 12

and explain the sign flip.

Solution set of −4x > 12; dividing by −4 flips the inequality

Example 4

medium

Solve

7 - 2x \geq 1

Solution set of 7 − 2x ≥ 1; dividing by −2 flips to ≤

Example 5

medium

Solve

\frac{2x + 1}{3} \geq 5

Example 6

hard

Solve

x^2 - x - 6 \leq 0

Example 7

challenge

Find all real

x

for which

\sqrt{x + 6} > x

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Solve

x - 4 < 10

Solution set of x − 4 < 10 graphed on the number line

Example 2

hard

Solve

4 - 2x \geq 10

Solution set of 4 − 2x ≥ 10 graphed on the number line

Example 3

easy

Solve:

x + 4 < 9

Solution set of x + 4 < 9

Example 4

easy

Solve:

x - 3 \ge 2

Solution set of x − 3 ≥ 2; closed circle shows 5 is included

Example 5

easy

Solve:

2x < 10

Solution set of 2x < 10; dividing by positive 2 keeps the direction

Example 6

easy

Solve:

-x < 3

Solution set of −x < 3 after flipping; open circle because strictly greater than

Example 7

easy

x = 4

a solution to

x > 4

Example 8

easy

Graph the solution to

x \le 2

on a number line — open or closed circle at

2

x ≤ 2 graphed: closed circle at 2 because ≤ includes the endpoint

Example 9

easy

Write 'all numbers greater than

-1

' as an inequality.

Example 10

easy

Solve:

3x + 1 > 7

Solution set of 3x + 1 > 7

Example 11

medium

Solve:

-2x + 5 \le 1

Solution set of −2x + 5 ≤ 1; flip when dividing by −2

Example 12

medium

Solve the compound inequality:

-3 < 2x - 1 \le 5

Solution set of −3 < 2x − 1 ≤ 5; left endpoint excluded, right included

Example 13

medium

For how many integers

x

-2 \le x < 3

true?

Example 14

medium

A taxi costs $3 plus $2 per mile. For what mileage

m

is the cost under $15?

Example 15

medium

Solve:

\frac{x}{-3} \ge 2

Example 16

medium

Solve:

2(x - 1) > x + 3

Example 17

medium

Solve:

5 - 3x < 2x - 10

Example 18

medium

Write the solution

x > 2

in interval notation.

Example 19

medium

a < b

, what is the relationship between

-a

and

-b

Example 20

challenge

Find all

x

with

|x - 3| < 5

and express in interval notation.

Solution set of |x − 3| < 5; both endpoints excluded

Example 21

challenge

Solve

x^2 < 9

and write the solution set.

Solution set of x² < 9; symmetric about 0, both endpoints excluded

Example 22

challenge

For what values of

k

does

kx > 6

give

x < 6/k

Example 23

easy

Solve

x + 7 \leq 12

Solution set of x + 7 ≤ 12

Example 24

easy

Solve

5x \geq 20

Example 25

easy

x = -2

a solution of

3x + 1 \geq -5

Example 26

easy

Solve

\frac{x}{4} < 3

Solution set of x/4 < 3

Example 27

medium

Solve

3(x - 2) \leq 9

Example 28

medium

Solve

4x - 5 < 2x + 7

Example 29

medium

Solve the compound inequality

1 \leq 3x - 2 < 10

Example 30

medium

Maya saves $8 per week. She already has $20. For how many weeks

w

will her total be at least $60?

Example 31

medium

Solve

-(x + 3) > 2x - 9

Example 32

medium

A rectangle has length

x + 4

and width

5

. Its perimeter is at most

30

. Find the largest integer value of

x

Example 33

hard

Solve

|2x - 1| \leq 7

and write the result in interval notation.

Example 34

hard

Solve

|3 - x| > 4

Example 35

hard

Solve

\frac{x - 1}{x + 2} \geq 0

Example 36

hard

For what values of

k

does the inequality

x^2 + kx + 4 > 0

hold for all real

x

Example 37

medium

A car rental costs $25 per day plus $0.20 per mile. With $100 for a 2-day rental, how many miles

m

can you drive?

Example 38

challenge

Find all real

x

with

\frac{1}{x} < 2

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Related Concepts

Equations Integers Absolute Value Linear Programming

Background Knowledge

These ideas may be useful before you work through the harder examples.

equationsintegers