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Solution - Absolute value equations

Exact form: x=1
x=1
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

|x|+|x−2|=0

Add −|x−2| to both sides of the equation:

|x|+|x−2|−|x−2|=−|x−2|

Simplify the arithmetic

|x|=−|x−2|

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
|x|=−|x−2|
without the absolute value bars:

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

3. Solve the two equations for x

7 additional steps

x=-(x-2)

Expand the parentheses:

x=−x+2

Add to both sides:

x+x=(-x+2)+x

Simplify the arithmetic:

2x=(-x+2)+x

Group like terms:

2x=(-x+x)+2

Simplify the arithmetic:

2x=2

Divide both sides by :

(2x)2=22

Simplify the fraction:

x=22

Simplify the fraction:

x=1

5 additional steps

x=-(-(x-2))

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

x=x−2

Subtract from both sides:

x-x=(x-2)-x

Simplify the arithmetic:

0=(x-2)-x

Group like terms:

0=(x-x)-2

Simplify the arithmetic:

0=−2

The statement is false:

0=−2

The equation is false so it has no solution.

4. List the solutions

x=1
(1 solution(s))

5. Graph

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