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Solution - Absolute value equations

Exact form: x=4,-83
x=4 , -\frac{8}{3}
Mixed number form: x=4,-223
x=4 , -2\frac{2}{3}
Decimal form: x=4,2.667
x=4 , -2.667

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+6|=2|x+1|
without the absolute value bars:

|x|=|y||x+6|=2|x+1|
x=+y(x+6)=2(x+1)
x=y(x+6)=2((x+1))
+x=y(x+6)=2(x+1)
x=y(x+6)=2(x+1)

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+6|=2|x+1|
x=+y , +x=y(x+6)=2(x+1)
x=y , x=y(x+6)=2((x+1))

2. Solve the two equations for x

12 additional steps

(x+6)=2·(x+1)

Expand the parentheses:

(x+6)=2x+2·1

Simplify the arithmetic:

(x+6)=2x+2

Subtract from both sides:

(x+6)-2x=(2x+2)-2x

Group like terms:

(x-2x)+6=(2x+2)-2x

Simplify the arithmetic:

-x+6=(2x+2)-2x

Group like terms:

-x+6=(2x-2x)+2

Simplify the arithmetic:

x+6=2

Subtract from both sides:

(-x+6)-6=2-6

Simplify the arithmetic:

x=26

Simplify the arithmetic:

x=4

Multiply both sides by :

-x·-1=-4·-1

Remove the one(s):

x=-4·-1

Simplify the arithmetic:

x=4

14 additional steps

(x+6)=2·(-(x+1))

Expand the parentheses:

(x+6)=2·(-x-1)

(x+6)=2·-x+2·-1

Group like terms:

(x+6)=(2·-1)x+2·-1

Multiply the coefficients:

(x+6)=-2x+2·-1

Simplify the arithmetic:

(x+6)=-2x-2

Add to both sides:

(x+6)+2x=(-2x-2)+2x

Group like terms:

(x+2x)+6=(-2x-2)+2x

Simplify the arithmetic:

3x+6=(-2x-2)+2x

Group like terms:

3x+6=(-2x+2x)-2

Simplify the arithmetic:

3x+6=2

Subtract from both sides:

(3x+6)-6=-2-6

Simplify the arithmetic:

3x=26

Simplify the arithmetic:

3x=8

Divide both sides by :

(3x)3=-83

Simplify the fraction:

x=-83

3. List the solutions

x=4,-83
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|x+6|
y=2|x+1|
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.