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Solution - Absolute value equations

Exact form: z=-2,23
z=-2 , \frac{2}{3}
Decimal form: z=−2,0.667
z=-2 , 0.667
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|=|z−2|
without the absolute value bars:

|x|=|y|2|z|=|z−2|
x=+y2(z)=(z−2)
x=−y2(z)=−(z−2)
+x=y2(z)=(z−2)
−x=y2(−(z))=(z−2)

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|=|z−2|
x=+y , +x=y2(z)=(z−2)
x=−y , −x=y2(z)=−(z−2)

2. Solve the two equations for z

3 additional steps

2z=(z-2)

Subtract from both sides:

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

Simplify the arithmetic:

z=(z-2)-z

Group like terms:

z=(z-z)-2

Simplify the arithmetic:

z=−2

6 additional steps

2z=-(z-2)

Expand the parentheses:

2z=−z+2

Add to both sides:

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

Simplify the arithmetic:

3z=(-z+2)+z

Group like terms:

3z=(-z+z)+2

Simplify the arithmetic:

3z=2

Divide both sides by :

(3z)3=23

Simplify the fraction:

z=23

3. List the solutions

z=-2,23
(2 solution(s))

4. Graph

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