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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 without absolute value bars

Use the rules:
|x|=|y| → x=±y and |x|=|y| → ±x=y
to write all four options of the equation
|2z|=|z|
without the absolute value bars:

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

2. Solve the two equations for z

2 additional steps

2z=z

Subtract from both sides:

(2z)-z=z-z

Simplify the arithmetic:

z=z−z

Simplify the arithmetic:

z=0

3 additional steps

2z=−z

Add to both sides:

(2z)+z=-z+z

Simplify the arithmetic:

3z=−z+z

Simplify the arithmetic:

3z=0

Divide both sides by the coefficient:

z=0

3. List the solutions

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

4. Graph

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