Solution Concept Examples in Math

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

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

Concept Recap

A specific value (or set of values) that makes an equation or inequality true when substituted in for the variable.

The answer to 'what value of $x$ makes this equation true?' — found by solving, confirmed by checking.

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 is a value that, substituted in, makes an equation or inequality a true statement.

Common stuck point: The procedure for solution concept is the easy part; the trap is skipping the check. Asking "Does plugging this value back into the original statement make it true?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does plugging this value back into the original statement make it true?

Worked Examples

Example 1

easy

x = 3

a solution of

x^2 - 9 = 0

Answer

Yes,

x = 3

is a solution.

First step

Substitute

x = 3

(3)^2 - 9 = 9 - 9 = 0

Full solution

2
Since the result is 0, $x = 3$ satisfies the equation.
3
Therefore $x = 3$ is a solution.

A solution is a value that, when substituted into the equation, makes the equation true. The verification process is called checking the solution.

Example 2

medium

Find all solutions of

|x| = 5

Example 3

medium

x=8

a solution of

3(x-1)=2x+5

Example 4

medium

Solve

|x-2|=7

Example 5

medium

Solve

x^2-5x+6=0

by factoring.

Example 6

hard

Solve

x^2=2x+15

Example 7

hard

Verify whether

x=4

is a solution of

\sqrt{x+5}=x-1

, including extraneous-root check.

Example 8

hard

How many solutions does

3x+1=3x+5

have?

Example 9

hard

Solve

x^2-6x+9=0

Example 10

challenge

Find all integer solutions of

x^2-y^2=15

with

x,y\ge 0

Practice Problems

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

Example 1

easy

Which values are solutions of

x^2 = 16

x = 4

x = -4

x = 8

Example 2

medium

Does the equation

x + 1 = x + 2

have any solutions?

Example 3

easy

x=4

a solution of

x+3=7

Example 4

easy

x=2

a solution of

3x=9

Example 5

easy

Solve

x-5=0

Example 6

easy

Solve

2x=8

and check.

Example 7

easy

Which value is a solution of

x+1=5

x=3

x=4

Example 8

easy

Solve

x/3=2

Example 9

easy

x=0

a solution of

5x=0

Example 10

easy

Solve

x+7=7

Example 11

medium

Solve

2x+1=9

and verify your solution.

Example 12

medium

Find all solutions of

x^2=4

Example 13

medium

A student solving

2(x-3)=4

wrote

x=2

. Verify whether this is correct.

Example 14

medium

Solve

\frac{x+2}{3}=4

and confirm.

Example 15

medium

x=-3

a solution of

x^2+x-6=0

? Substitute to decide.

Example 16

medium

Solve

5-2x=1

and verify.

Example 17

challenge

Find all solutions of

|x-2|=5

and verify each.

Example 18

challenge

The equation

\sqrt{x}=x-2

gives candidates

x=1

and

x=4

after squaring. Determine which are actual solutions.

Example 19

challenge

For the equation

x^2-6x+9=0

, find the solution and explain why it counts as one value.

Example 20

medium

Solve

3x-4=2x+1

and verify.

Example 21

medium

x=-1

a solution of

x^2+3x+2=0

Example 22

medium

Solve

\frac{x}{2}-1=3

and verify.

Example 23

easy

x=2

a solution of

x^2-3x+2=0

Example 24

easy

Which of

x=-2, x=0, x=2

are solutions of

x^2=4

Example 25

easy

x=-1

a solution of

2x+3=1

Example 26

easy

Does

x=3

satisfy

2x-5>0

Example 27

medium

Solve

3x+4=2x-5

Example 28

medium

Find all solutions of

(x-3)(x+4)=0

Example 29

medium

For what

x

does

\frac{1}{x-3}=2

hold?

Example 30

medium

x=4

a solution of

\sqrt{x+5}=3

Example 31

medium

Solve

2(x+3)=x+11

Example 32

hard

Find all real

x

with

\frac{2x}{x-1}=4

Example 33

hard

Solve

|2x-1|=5

Example 34

hard

How many solutions does

2(x+3)=2x+6

have?

Example 35

hard

For what value of

k

x=2

a solution of

kx-3=5

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Related Concepts

Equations Solution Set Checking Solutions

Background Knowledge

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

equations