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Solution - Absolute value equations

Exact form: =6,−2
=6 , -2
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|=|z−2|
without the absolute value bars:

|x|=|y||+4|=|z−2|
x=+y(+4)=(z−2)
x=−y(+4)=−(z−2)
+x=y(+4)=(z−2)
−x=y−(+4)=(z−2)

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|=|z−2|
x=+y , +x=y(+4)=(z−2)
x=−y , −x=y(+4)=−(z−2)

2. Solve the two equations for

3 additional steps

(4)=(z-2)

Swap sides:

(z-2)=(4)

Add to both sides:

(z-2)+2=(4)+2

Simplify the arithmetic:

z=(4)+2

Simplify the arithmetic:

z=6

7 additional steps

(4)=-(z-2)

Expand the parentheses:

(4)=-z+2

Swap sides:

-z+2=(4)

Subtract from both sides:

(-z+2)-2=(4)-2

Simplify the arithmetic:

-z=(4)-2

Simplify the arithmetic:

−z=2

Multiply both sides by :

-z·-1=2·-1

Remove the one(s):

z=2·-1

Simplify the arithmetic:

z=−2

3. List the solutions

=6,−2
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|+4|
y=|z−2|
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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