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Solution - Absolute value equations

Exact form: x=-18,-125
x=-18 , -\frac{12}{5}
Mixed number form: x=-18,-225
x=-18 , -2\frac{2}{5}
Decimal form: x=18,2.4
x=-18 , -2.4

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
|2x3|=3|x+5|
without the absolute value bars:

|x|=|y||2x3|=3|x+5|
x=+y(2x3)=3(x+5)
x=y(2x3)=3((x+5))
+x=y(2x3)=3(x+5)
x=y(2x3)=3(x+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||2x3|=3|x+5|
x=+y , +x=y(2x3)=3(x+5)
x=y , x=y(2x3)=3((x+5))

2. Solve the two equations for x

12 additional steps

(2x-3)=3·(x+5)

Expand the parentheses:

(2x-3)=3x+3·5

Simplify the arithmetic:

(2x-3)=3x+15

Subtract from both sides:

(2x-3)-3x=(3x+15)-3x

Group like terms:

(2x-3x)-3=(3x+15)-3x

Simplify the arithmetic:

-x-3=(3x+15)-3x

Group like terms:

-x-3=(3x-3x)+15

Simplify the arithmetic:

x3=15

Add to both sides:

(-x-3)+3=15+3

Simplify the arithmetic:

x=15+3

Simplify the arithmetic:

x=18

Multiply both sides by :

-x·-1=18·-1

Remove the one(s):

x=18·-1

Simplify the arithmetic:

x=18

14 additional steps

(2x-3)=3·(-(x+5))

Expand the parentheses:

(2x-3)=3·(-x-5)

(2x-3)=3·-x+3·-5

Group like terms:

(2x-3)=(3·-1)x+3·-5

Multiply the coefficients:

(2x-3)=-3x+3·-5

Simplify the arithmetic:

(2x-3)=-3x-15

Add to both sides:

(2x-3)+3x=(-3x-15)+3x

Group like terms:

(2x+3x)-3=(-3x-15)+3x

Simplify the arithmetic:

5x-3=(-3x-15)+3x

Group like terms:

5x-3=(-3x+3x)-15

Simplify the arithmetic:

5x3=15

Add to both sides:

(5x-3)+3=-15+3

Simplify the arithmetic:

5x=15+3

Simplify the arithmetic:

5x=12

Divide both sides by :

(5x)5=-125

Simplify the fraction:

x=-125

3. List the solutions

x=-18,-125
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|2x3|
y=3|x+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.