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Solution - Absolute value equations

Exact form: =12,20
=12 , 20
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
|−4|=|x−16|
without the absolute value bars:

|x|=|y||−4|=|x−16|
x=+y(−4)=(x−16)
x=−y(−4)=−(x−16)
+x=y(−4)=(x−16)
−x=y−(−4)=(x−16)

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||−4|=|x−16|
x=+y , +x=y(−4)=(x−16)
x=−y , −x=y(−4)=−(x−16)

2. Solve the two equations for

3 additional steps

-4=(x-16)

Swap sides:

(x-16)=-4

Add to both sides:

(x-16)+16=-4+16

Simplify the arithmetic:

x=−4+16

Simplify the arithmetic:

x=12

7 additional steps

-4=-(x-16)

Expand the parentheses:

−4=−x+16

Swap sides:

−x+16=−4

Subtract from both sides:

(-x+16)-16=-4-16

Simplify the arithmetic:

−x=−4−16

Simplify the arithmetic:

−x=−20

Multiply both sides by :

-x·-1=-20·-1

Remove the one(s):

x=-20·-1

Simplify the arithmetic:

x=20

3. List the solutions

=12,20
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|−4|
y=|x−16|
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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