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Solution - Absolute value equations

Exact form: =0,−4
=0 , -4
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
|+2|=|y+2|
without the absolute value bars:

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

2. Solve the two equations for

3 additional steps

(2)=(y+2)

Swap sides:

(y+2)=(2)

Subtract from both sides:

(y+2)-2=(2)-2

Simplify the arithmetic:

y=(2)-2

Simplify the arithmetic:

y=0

7 additional steps

(2)=-(y+2)

Expand the parentheses:

(2)=-y-2

Swap sides:

-y-2=(2)

Add to both sides:

(-y-2)+2=(2)+2

Simplify the arithmetic:

-y=(2)+2

Simplify the arithmetic:

−y=4

Multiply both sides by :

-y·-1=4·-1

Remove the one(s):

y=4·-1

Simplify the arithmetic:

y=−4

3. List the solutions

=0,−4
(2 solution(s))

4. Graph

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