Solution Set Examples: 42 Problems with Answers

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

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

Concept Recap

The complete set of all values that satisfy a given equation or inequality — it may be empty, finite, or infinite.

Not just one answer, but ALL answers that work — an inequality like $x > 3$ has infinitely many.

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: A solution set is the complete collection of all values that satisfy an equation or inequality.

Common stuck point: The procedure for solution set is the easy part; the trap is giving one value when there are many. Asking "Am I describing EVERY value that satisfies the statement, not just one?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I describing EVERY value that satisfies the statement, not just one?

Worked Examples

Example 1

easy

What is the solution set of

x^2 = 25

Answer

{- 5, 5}

First step

Find all values where

x^2 = 25

x = 5

x = -5

Full solution

2
Write as a set: $\{5, -5\}$ .
3
The solution set contains every value that satisfies the equation.

A solution set is the collection of all values that make the equation true. It can contain zero, one, two, or infinitely many elements.

Example 2

medium

What is the solution set of

x + 3 > 5

Example 3

medium

Find the solution set of

x^2-7x+12=0

Example 4

medium

Find the solution set of

|x|<3

Example 5

medium

Express the solution set of

1\le x<6

in interval and set-builder form.

Example 6

hard

Find the solution set of

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

Example 7

hard

Find the solution set of

|x-3|\le 4

Example 8

hard

Find the solution set of

\frac{1}{x}>2

for

x>0

Example 9

hard

Find the solution set of

(x-1)(x+2)>0

Example 10

challenge

Find the solution set of

|x-1|+|x+2|=5

Practice Problems

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

Example 1

easy

What is the solution set of

2x = 10

Example 2

medium

What is the solution set of

x^2 + 1 = 0

over the real numbers?

Example 3

easy

Write the solution set of

x=5

Example 4

easy

Write the solution set of

x^2=9

Example 5

easy

Describe the solution set of

x>3

Example 6

easy

What is the solution set of

0\cdot x=5

Example 7

easy

What is the solution set of

0\cdot x=0

Example 8

easy

Write the solution set of

x+2=2

Example 9

easy

Describe the solution set of

x\le 0

Example 10

easy

Solution set of

x^2=-1

over the reals?

Example 11

medium

Find the solution set of

x^2-x-6=0

Example 12

medium

Solve the inequality

2x-1<5

and write the solution set.

Example 13

medium

Write the solution set of

|x|=4

Example 14

medium

Solve

-3x\ge 6

and write the solution set.

Example 15

medium

Find the solution set of

x^2=2x

Example 16

medium

Solve the compound condition

-1\le x<3

and describe its solution set.

Example 17

challenge

Find the solution set of

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

, noting any excluded value.

Example 18

challenge

Find the solution set of

x^3=x

Example 19

challenge

For what values of

k

does

x^2=k

have a solution set with exactly one element (over the reals)?

Example 20

medium

Write the solution set of

3x=12

Example 21

medium

Find the solution set of

x^2-4=0

Example 22

medium

Solve

x+5>5

and write the solution set.

Example 23

easy

Write the solution set of

(x-1)(x-4)=0

Example 24

easy

Write the solution set of

x\ge -2

in interval notation.

Example 25

easy

Write the solution set of

x^2=0

Example 26

medium

Write the solution set of

2x-1\ge 7

in interval notation.

Example 27

medium

Find the solution set of

|x|>2

Example 28

medium

What is the solution set of

3x+5=3x+7

Example 29

medium

Find the solution set of

x(x-2)(x+5)=0

Example 30

hard

Find the solution set of

x^2-5x+6\le 0

Example 31

hard

Find the solution set of

\sqrt{x}=x-2

over the reals.

Example 32

hard

Write the solution set of the system

x+y=5

x-y=1

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Related Concepts

Solution Concept Interval Notation Empty Set

Background Knowledge

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

solution concept