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Solution - Absolute value equations

Exact form: x=-103,-12
x=-\frac{10}{3} , -\frac{1}{2}
Mixed number form: x=-313,-12
x=-3\frac{1}{3} , -\frac{1}{2}
Decimal form: x=3.333,0.5
x=-3.333 , -0.5

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
|7x+12|=|x8|
without the absolute value bars:

|x|=|y||7x+12|=|x8|
x=+y(7x+12)=(x8)
x=y(7x+12)=(x8)
+x=y(7x+12)=(x8)
x=y(7x+12)=(x8)

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||7x+12|=|x8|
x=+y , +x=y(7x+12)=(x8)
x=y , x=y(7x+12)=(x8)

2. Solve the two equations for x

11 additional steps

(7x+12)=(x-8)

Subtract from both sides:

(7x+12)-x=(x-8)-x

Group like terms:

(7x-x)+12=(x-8)-x

Simplify the arithmetic:

6x+12=(x-8)-x

Group like terms:

6x+12=(x-x)-8

Simplify the arithmetic:

6x+12=8

Subtract from both sides:

(6x+12)-12=-8-12

Simplify the arithmetic:

6x=812

Simplify the arithmetic:

6x=20

Divide both sides by :

(6x)6=-206

Simplify the fraction:

x=-206

Find the greatest common factor of the numerator and denominator:

x=(-10·2)(3·2)

Factor out and cancel the greatest common factor:

x=-103

12 additional steps

(7x+12)=-(x-8)

Expand the parentheses:

(7x+12)=-x+8

Add to both sides:

(7x+12)+x=(-x+8)+x

Group like terms:

(7x+x)+12=(-x+8)+x

Simplify the arithmetic:

8x+12=(-x+8)+x

Group like terms:

8x+12=(-x+x)+8

Simplify the arithmetic:

8x+12=8

Subtract from both sides:

(8x+12)-12=8-12

Simplify the arithmetic:

8x=812

Simplify the arithmetic:

8x=4

Divide both sides by :

(8x)8=-48

Simplify the fraction:

x=-48

Find the greatest common factor of the numerator and denominator:

x=(-1·4)(2·4)

Factor out and cancel the greatest common factor:

x=-12

3. List the solutions

x=-103,-12
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|7x+12|
y=|x8|
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.