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Solution - Absolute value equations

Exact form: a=0
a=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

|a−2|+|a+2|=0

Add −|a+2| to both sides of the equation:

|a−2|+|a+2|−|a+2|=−|a+2|

Simplify the arithmetic

|a−2|=−|a+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
|a−2|=−|a+2|
without the absolute value bars:

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

3. Solve the two equations for a

9 additional steps

(a-2)=-(a+2)

Expand the parentheses:

(a-2)=-a-2

Add to both sides:

(a-2)+a=(-a-2)+a

Group like terms:

(a+a)-2=(-a-2)+a

Simplify the arithmetic:

2a-2=(-a-2)+a

Group like terms:

2a-2=(-a+a)-2

Simplify the arithmetic:

2a−2=−2

Add to both sides:

(2a-2)+2=-2+2

Simplify the arithmetic:

2a=−2+2

Simplify the arithmetic:

2a=0

Divide both sides by the coefficient:

a=0

6 additional steps

(a-2)=-(-(a+2))

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

(a-2)=a+2

Subtract from both sides:

(a-2)-a=(a+2)-a

Group like terms:

(a-a)-2=(a+2)-a

Simplify the arithmetic:

-2=(a+2)-a

Group like terms:

-2=(a-a)+2

Simplify the arithmetic:

−2=2

The statement is false:

−2=2

The equation is false so it has no solution.

4. List the solutions

a=0
(1 solution(s))

5. Graph

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