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Solution - Absolute value equations

Exact form: z=12
z=\frac{1}{2}
Decimal form: z=0.5
z=0.5
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−1|=0

Add −|z−1| to both sides of the equation:

|z|+|z−1|−|z−1|=−|z−1|

Simplify the arithmetic

|z|=−|z−1|

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−1|
without the absolute value bars:

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

3. Solve the two equations for z

6 additional steps

z=-(z-1)

Expand the parentheses:

z=−z+1

Add to both sides:

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

Simplify the arithmetic:

2z=(-z+1)+z

Group like terms:

2z=(-z+z)+1

Simplify the arithmetic:

2z=1

Divide both sides by :

(2z)2=12

Simplify the fraction:

z=12

5 additional steps

z=-(-(z-1))

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

z=z−1

Subtract from both sides:

z-z=(z-1)-z

Simplify the arithmetic:

0=(z-1)-z

Group like terms:

0=(z-z)-1

Simplify the arithmetic:

0=−1

The statement is false:

0=−1

The equation is false so it has no solution.

4. List the solutions

z=12
(1 solution(s))

5. Graph

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