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

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

2. Solve the two equations for z

5 additional steps

z=2z

Subtract from both sides:

z-2z=(2z)-2z

Simplify the arithmetic:

-z=(2z)-2z

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

5 additional steps

z=2·-z

Group like terms:

z=(2·-1)z

Multiply the coefficients:

z=−2z

Add to both sides:

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

Simplify the arithmetic:

3z=(-2z)+2z

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=|z|
y=2|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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