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Solution - Absolute value equations

Exact form: x=0
x=0
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+1|+|x−1|=0

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

|x+1|+|x−1|−|x−1|=−|x−1|

Simplify the arithmetic

|x+1|=−|x−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
|x+1|=−|x−1|
without the absolute value bars:

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

3. Solve the two equations for x

9 additional steps

(x+1)=-(x-1)

Expand the parentheses:

(x+1)=-x+1

Add to both sides:

(x+1)+x=(-x+1)+x

Group like terms:

(x+x)+1=(-x+1)+x

Simplify the arithmetic:

2x+1=(-x+1)+x

Group like terms:

2x+1=(-x+x)+1

Simplify the arithmetic:

2x+1=1

Subtract from both sides:

(2x+1)-1=1-1

Simplify the arithmetic:

2x=1−1

Simplify the arithmetic:

2x=0

Divide both sides by the coefficient:

x=0

6 additional steps

(x+1)=-(-(x-1))

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

(x+1)=x-1

Subtract from both sides:

(x+1)-x=(x-1)-x

Group like terms:

(x-x)+1=(x-1)-x

Simplify the arithmetic:

1=(x-1)-x

Group like terms:

1=(x-x)-1

Simplify the arithmetic:

1=−1

The statement is false:

1=−1

The equation is false so it has no solution.

4. List the solutions

x=0
(1 solution(s))

5. Graph

Each line represents the function of one side of the equation:
y=|x+1|
y=−|x−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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