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Solution - Absolute value equations

Exact form: x=-12
x=-\frac{1}{2}
Decimal form: x=−0.5
x=-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

|x+3|+|−x+2|=0

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

|x+3|+|−x+2|−|−x+2|=−|−x+2|

Simplify the arithmetic

|x+3|=−|−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+3|=−|−x+2|
without the absolute value bars:

|x|=|y||x+3|=−|−x+2|
x=+y(x+3)=−(−x+2)
x=−y(x+3)=−−(−x+2)
+x=y(x+3)=−(−x+2)
−x=y−(x+3)=−(−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+3|=−|−x+2|
x=+y , +x=y(x+3)=−(−x+2)
x=−y , −x=y(x+3)=−−(−x+2)

3. Solve the two equations for x

6 additional steps

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

Expand the parentheses:

(x+3)=x-2

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

3=(x-2)-x

Group like terms:

3=(x-x)-2

Simplify the arithmetic:

3=−2

The statement is false:

3=−2

The equation is false so it has no solution.

10 additional steps

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

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

(x+3)=-x+2

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

2x+3=2

Subtract from both sides:

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

Simplify the arithmetic:

2x=2−3

Simplify the arithmetic:

2x=−1

Divide both sides by :

(2x)2=-12

Simplify the fraction:

x=-12

4. Graph

Each line represents the function of one side of the equation:
y=|x+3|
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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