Inequality Intuition Examples in Math

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

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

Concept Recap

Understanding that $<$ and $>$ describe ordering relationships—one quantity is strictly smaller or larger than the other.

If $5 < 7$ , then 5 is somewhere to the left of 7 on the number line.

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 with $<$ or $>$ says one quantity is definitely smaller or larger, fixing their order on the number line.

Common stuck point: The procedure for inequality intuition is the easy part; the trap is forgetting to flip the symbol when multiplying or dividing by a negative. Asking "Does the statement order two quantities (strictly smaller or larger) rather than equate them?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the statement order two quantities (strictly smaller or larger) rather than equate them?

Worked Examples

Example 1

easy

Solve

x + 3 < 10

and graph the solution on a number line.

Answer

x < 7

First step

Subtract 3 from both sides:

x < 7

Full solution

2
Solution: all numbers less than 7.
3
Graph: open circle at 7, arrow pointing left.
4
Example values: $x = 6, 5, 0, -1, \ldots$

Inequalities are solved like equations but with a direction. The solution is a range of values, not a single point. Open circle means 7 is not included.

Example 2

medium

Solve

-2x \geq 8

and explain the direction flip when multiplying by a negative.

Example 3

easy

Solve

x - 5 \geq -2

and show on a number line.

Example 4

medium

Explain why

a < b

implies

a + c < b + c

for any real

c

Example 5

medium

Show on a number line:

x > -1

and

x \leq 4

. Write the intersection.

Example 6

challenge

Prove: if

a, b > 0

, then

a + b \geq 2\sqrt{ab}

Practice Problems

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

Example 1

easy

Solve

2x - 4 > 6

Example 2

medium

A student must score at least 70 to pass. They have scores 65, 72, 80. Which scores pass? Write as inequalities.

Example 3

easy

Which symbol makes it true:

5 \ \square\ 8

Example 4

easy

Read

3 > x

in words.

Example 5

easy

Does

x \le 5

include the value

5

Example 6

easy

Does

x < 5

include the value

5

Example 7

easy

On a number line, is

-3

to the left or right of

2

Example 8

easy

Which is true:

-5 < -2

-5 > -2

Example 9

easy

x \ge 0

satisfied by

x = 0

Example 10

easy

Rewrite

7 > x

with the variable on the left.

Example 11

medium

Solve

x + 3 < 10

Example 12

medium

Solve

-2x > 6

Example 13

medium

Solve

3x - 1 \ge 8

Example 14

medium

Which integers satisfy

-1 < x \le 3

Example 15

medium

Solve

5 - x \le 2

Example 16

medium

A ride requires height

\ge 48

inches. Write the inequality and state if

48

inches qualifies.

Example 17

medium

Order from least to greatest:

-\frac{3}{2},\ 0.5,\ -1,\ 2

Example 18

challenge

Solve

-3 < 2x + 1 \le 7

and give the integer solutions.

Example 19

challenge

For which values of

x

-2x + 1 > 5

? Explain the sign flip.

Example 20

challenge

Explain why multiplying both sides of

2 < 4

-1

gives

-2 > -4

, not

-2 < -4

Example 21

medium

Solve

\frac{x}{2} + 1 < 4

Example 22

medium

A number

n

satisfies 'at most

10

and more than

4

'. Write the compound inequality.

Example 23

easy

True or false:

x \geq 4

is satisfied by

x = 4

Example 24

easy

List the integers that satisfy

-2 \leq x < 2

Example 25

easy

Order from least to greatest:

\frac{1}{2}, -\frac{3}{4}, 0, -1

Example 26

easy

Translate: 'a price

p

no more than $50' as an inequality.

Example 27

medium

Solve

-3x \leq 9

Example 28

medium

Solve

2(x+1) \leq 3x - 4

Example 29

medium

a > b

and

b > c

, what can you conclude about

a

and

c

Example 30

medium

A car uses up to

12

gallons per trip and contains

g

gallons. Write an inequality for safe fuel.

Example 31

medium

Which of the following violates

|x| < 3

x = 2.9

x = 3

x = -2

Example 32

medium

Solve

\frac{x}{-2} > 4

Example 33

medium

True or false: if

a < b

then

a^2 < b^2

Example 34

medium

How many positive integers

n

satisfy

n < 7

Example 35

hard

Solve the compound inequality

-5 < 1 - 2x \leq 3

Example 36

hard

-2 < x < 5

, find the range of

3 - 2x

Example 37

hard

For which positive integers

n

does

\frac{1}{n} > \frac{1}{10}

Example 38

hard

A student claims: 'If

a < b

then

\frac{1}{a} > \frac{1}{b}

.' Give a counterexample.

Example 39

challenge

For which real

x

\frac{x}{x - 2} \leq 1

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Related Concepts

More and Less Comparison Inequalities Solving Linear Equations

Background Knowledge

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

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