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Solution - Absolute value equations

Exact form: x=-23,-23
x=-\frac{2}{3} , -\frac{2}{3}
Decimal form: x=−0.667,−0.667
x=-0.667 , -0.667
See steps

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. 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+23|=0|-x+8|
without the absolute value bars:

|x|=|y||x+23|=0|-x+8|
x=+y(x+23)=0(-x+8)
x=-y(x+23)=0(-(-x+8))
+x=y(x+23)=0(-x+8)
-x=y-(x+23)=0(-x+8)

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+23|=0|-x+8|
x=+y , +x=y(x+23)=0(-x+8)
x=-y , -x=y(x+23)=0(-(-x+8))

2. Solve the two equations for x

6 additional steps

(x+23)=0·(-x+8)

NT_MSLUS_MAINSTEP_MULTIPLY_BY_ZERO:

(x+23)=0

Subtract from both sides:

(x+23)-23=0-23

Combine the fractions:

x+(2-2)3=0-23

Combine the numerators:

x+03=0-23

Reduce the zero numerator:

x+0=0-23

Simplify the arithmetic:

x=0-23

Simplify the arithmetic:

x=-23

6 additional steps

(x+23)=0·(-(-x+8))

NT_MSLUS_MAINSTEP_MULTIPLY_BY_ZERO:

(x+23)=0

Subtract from both sides:

(x+23)-23=0-23

Combine the fractions:

x+(2-2)3=0-23

Combine the numerators:

x+03=0-23

Reduce the zero numerator:

x+0=0-23

Simplify the arithmetic:

x=0-23

Simplify the arithmetic:

x=-23

3. List the solutions

x=-23,-23
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|x+23|
y=0|-x+8|
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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