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Solution - Absolute value equations

Exact form: x=7.5,0.9
x=7.5 , -0.9

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
|2x+6|=3|x0.5|
without the absolute value bars:

|x|=|y||2x+6|=3|x0.5|
x=+y(2x+6)=3(x0.5)
x=y(2x+6)=3((x0.5))
+x=y(2x+6)=3(x0.5)
x=y(2x+6)=3(x0.5)

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||2x+6|=3|x0.5|
x=+y , +x=y(2x+6)=3(x0.5)
x=y , x=y(2x+6)=3((x0.5))

2. Solve the two equations for x

12 additional steps

(2x+6)=3·(x-0.5)

Expand the parentheses:

(2x+6)=3x+3·-0.5

Simplify the arithmetic:

(2x+6)=3x-1.5

Subtract from both sides:

(2x+6)-3x=(3x-1.5)-3x

Group like terms:

(2x-3x)+6=(3x-1.5)-3x

Simplify the arithmetic:

-x+6=(3x-1.5)-3x

Group like terms:

-x+6=(3x-3x)-1.5

Simplify the arithmetic:

x+6=1.5

Subtract from both sides:

(-x+6)-6=-1.5-6

Simplify the arithmetic:

x=1.56

Simplify the arithmetic:

x=7.5

Multiply both sides by :

-x·-1=-7.5·-1

Remove the one(s):

x=-7.5·-1

Simplify the arithmetic:

x=7.5

15 additional steps

(2x+6)=3·(-(x-0.5))

Expand the parentheses:

(2x+6)=3·(-x+0.5)

Expand the parentheses:

(2x+6)=3·-x+3·0.5

Group like terms:

(2x+6)=(3·-1)x+3·0.5

Multiply the coefficients:

(2x+6)=-3x+3·0.5

Simplify the arithmetic:

(2x+6)=-3x+1.5

Add to both sides:

(2x+6)+3x=(-3x+1.5)+3x

Group like terms:

(2x+3x)+6=(-3x+1.5)+3x

Simplify the arithmetic:

5x+6=(-3x+1.5)+3x

Group like terms:

5x+6=(-3x+3x)+1.5

Simplify the arithmetic:

5x+6=1.5

Subtract from both sides:

(5x+6)-6=1.5-6

Simplify the arithmetic:

5x=1.56

Simplify the arithmetic:

5x=4.5

Divide both sides by :

(5x)5=-4.55

Simplify the fraction:

x=-4.55

Simplify the arithmetic:

x=0.9

3. List the solutions

x=7.5,0.9
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|2x+6|
y=3|x0.5|
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.