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Solution - Absolute value equations

Exact form: =15,−3
=15 , -3
See steps

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
|x|=|y| → x=±y and |x|=|y| → ±x=y
to write all four options of the equation
|+9|=|x−6|
without the absolute value bars:

|x|=|y||+9|=|x−6|
x=+y(+9)=(x−6)
x=−y(+9)=−(x−6)
+x=y(+9)=(x−6)
−x=y−(+9)=(x−6)

When simplified, equations x=+y and +x=y are the same and equations x=−y and −x=y are the same, so we end up with only 2 equations:

|x|=|y||+9|=|x−6|
x=+y , +x=y(+9)=(x−6)
x=−y , −x=y(+9)=−(x−6)

2. Solve the two equations for

3 additional steps

(9)=(x-6)

Swap sides:

(x-6)=(9)

Add to both sides:

(x-6)+6=(9)+6

Simplify the arithmetic:

x=(9)+6

Simplify the arithmetic:

x=15

7 additional steps

(9)=-(x-6)

Expand the parentheses:

(9)=-x+6

Swap sides:

-x+6=(9)

Subtract from both sides:

(-x+6)-6=(9)-6

Simplify the arithmetic:

-x=(9)-6

Simplify the arithmetic:

−x=3

Multiply both sides by :

-x·-1=3·-1

Remove the one(s):

x=3·-1

Simplify the arithmetic:

x=−3

3. List the solutions

=15,−3
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|+9|
y=|x−6|
The equation is true where the two lines cross.

Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

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