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Solution - Absolute value equations

Exact form: =2,0
=2 , 0
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
|+1|=|z−1|
without the absolute value bars:

|x|=|y||+1|=|z−1|
x=+y(+1)=(z−1)
x=−y(+1)=−(z−1)
+x=y(+1)=(z−1)
−x=y−(+1)=(z−1)

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||+1|=|z−1|
x=+y , +x=y(+1)=(z−1)
x=−y , −x=y(+1)=−(z−1)

2. Solve the two equations for

3 additional steps

(1)=(z-1)

Swap sides:

(z-1)=(1)

Add to both sides:

(z-1)+1=(1)+1

Simplify the arithmetic:

z=(1)+1

Simplify the arithmetic:

z=2

7 additional steps

(1)=-(z-1)

Expand the parentheses:

(1)=-z+1

Swap sides:

-z+1=(1)

Subtract from both sides:

(-z+1)-1=(1)-1

Simplify the arithmetic:

-z=(1)-1

Simplify the arithmetic:

−z=0

Multiply both sides by :

-z·-1=0·-1

Remove the one(s):

z=0·-1

Multiply by zero:

z=0

3. List the solutions

=2,0
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|+1|
y=|z−1|
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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