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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−3|+|−x+3|=0

Add −|−x+3| to both sides of the equation:

|−x−3|+|−x+3|−|−x+3|=−|−x+3|

Simplify the arithmetic

|−x−3|=−|−x+3|

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−3|=−|−x+3|
without the absolute value bars:

|x|=|y||−x−3|=−|−x+3|
x=+y(−x−3)=−(−x+3)
x=−y(−x−3)=−−(−x+3)
+x=y(−x−3)=−(−x+3)
−x=y−(−x−3)=−(−x+3)

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−3|=−|−x+3|
x=+y , +x=y(−x−3)=−(−x+3)
x=−y , −x=y(−x−3)=−−(−x+3)

3. Solve the two equations for x

9 additional steps

(-x-3)=-(-x+3)

Expand the parentheses:

(-x-3)=x-3

Subtract from both sides:

(-x-3)-x=(x-3)-x

Group like terms:

(-x-x)-3=(x-3)-x

Simplify the arithmetic:

-2x-3=(x-3)-x

Group like terms:

-2x-3=(x-x)-3

Simplify the arithmetic:

−2x−3=−3

Add to both sides:

(-2x-3)+3=-3+3

Simplify the arithmetic:

−2x=−3+3

Simplify the arithmetic:

−2x=0

Divide both sides by the coefficient:

x=0

6 additional steps

(-x-3)=-(-(-x+3))

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

(-x-3)=-x+3

Add to both sides:

(-x-3)+x=(-x+3)+x

Group like terms:

(-x+x)-3=(-x+3)+x

Simplify the arithmetic:

-3=(-x+3)+x

Group like terms:

-3=(-x+x)+3

Simplify the arithmetic:

−3=3

The statement is false:

−3=3

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−3|
y=−|−x+3|
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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