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Solution - Absolute value equations

Exact form: x=-110
x=-\frac{1}{10}
Decimal form: x=0.1
x=-0.1

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
|5x8|=|5x+7|
without the absolute value bars:

|x|=|y||5x8|=|5x+7|
x=+y(5x8)=(5x+7)
x=y(5x8)=(5x+7)
+x=y(5x8)=(5x+7)
x=y(5x8)=(5x+7)

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||5x8|=|5x+7|
x=+y , +x=y(5x8)=(5x+7)
x=y , x=y(5x8)=(5x+7)

2. Solve the two equations for x

5 additional steps

(-5x-8)=(-5x+7)

Add to both sides:

(-5x-8)+5x=(-5x+7)+5x

Group like terms:

(-5x+5x)-8=(-5x+7)+5x

Simplify the arithmetic:

-8=(-5x+7)+5x

Group like terms:

-8=(-5x+5x)+7

Simplify the arithmetic:

8=7

The statement is false:

8=7

The equation is false so it has no solution.

12 additional steps

(-5x-8)=-(-5x+7)

Expand the parentheses:

(-5x-8)=5x-7

Subtract from both sides:

(-5x-8)-5x=(5x-7)-5x

Group like terms:

(-5x-5x)-8=(5x-7)-5x

Simplify the arithmetic:

-10x-8=(5x-7)-5x

Group like terms:

-10x-8=(5x-5x)-7

Simplify the arithmetic:

10x8=7

Add to both sides:

(-10x-8)+8=-7+8

Simplify the arithmetic:

10x=7+8

Simplify the arithmetic:

10x=1

Divide both sides by :

(-10x)-10=1-10

Cancel out the negatives:

10x10=1-10

Simplify the fraction:

x=1-10

Move the negative sign from the denominator to the numerator:

x=-110

3. Graph

Each line represents the function of one side of the equation:
y=|5x8|
y=|5x+7|
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.