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Solution - Absolute value equations

Exact form: =5,1
=5 , 1
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|=|z−3|
without the absolute value bars:

|x|=|y||+2|=|z−3|
x=+y(+2)=(z−3)
x=−y(+2)=−(z−3)
+x=y(+2)=(z−3)
−x=y−(+2)=(z−3)

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|=|z−3|
x=+y , +x=y(+2)=(z−3)
x=−y , −x=y(+2)=−(z−3)

2. Solve the two equations for

3 additional steps

(2)=(z-3)

Swap sides:

(z-3)=(2)

Add to both sides:

(z-3)+3=(2)+3

Simplify the arithmetic:

z=(2)+3

Simplify the arithmetic:

z=5

7 additional steps

(2)=-(z-3)

Expand the parentheses:

(2)=-z+3

Swap sides:

-z+3=(2)

Subtract from both sides:

(-z+3)-3=(2)-3

Simplify the arithmetic:

-z=(2)-3

Simplify the arithmetic:

−z=−1

Multiply both sides by :

-z·-1=-1·-1

Remove the one(s):

z=-1·-1

Simplify the arithmetic:

z=1

3. List the solutions

=5,1
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|+2|
y=|z−3|
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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