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Solution - Absolute value equations

Exact form: z=0,0
z=0 , 0
See steps

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

|z|+|z|=0

Add −|z| to both sides of the equation:

|z|+|z|−|z|=−|z|

Simplify the arithmetic

|z|=−|z|

2. 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
|z|=−|z|
without the absolute value bars:

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

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

3. Solve the two equations for z

3 additional steps

z=−z

Add to both sides:

z+z=−z+z

Simplify the arithmetic:

2z=−z+z

Simplify the arithmetic:

2z=0

Divide both sides by the coefficient:

z=0

2 additional steps

z=z

Subtract from both sides:

z−z=z−z

Simplify the arithmetic:

0=z−z

Simplify the arithmetic:

0=0

4. List the solutions

z=0,0
(2 solution(s))

5. Graph

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