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Solution - Absolute value equations

Exact form: x=5,3
x=5 , -3

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

|x9|+|2x6|=0

Add |2x6| to both sides of the equation:

|x9|+|2x6||2x6|=|2x6|

Simplify the arithmetic

|x9|=|2x6|

2. 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
|x9|=|2x6|
without the absolute value bars:

|x|=|y||x9|=|2x6|
x=+y(x9)=(2x6)
x=y(x9)=(2x6)
+x=y(x9)=(2x6)
x=y(x9)=(2x6)

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||x9|=|2x6|
x=+y , +x=y(x9)=(2x6)
x=y , x=y(x9)=(2x6)

3. Solve the two equations for x

12 additional steps

(x-9)=-(2x-6)

Expand the parentheses:

(x-9)=-2x+6

Add to both sides:

(x-9)+2x=(-2x+6)+2x

Group like terms:

(x+2x)-9=(-2x+6)+2x

Simplify the arithmetic:

3x-9=(-2x+6)+2x

Group like terms:

3x-9=(-2x+2x)+6

Simplify the arithmetic:

3x9=6

Add to both sides:

(3x-9)+9=6+9

Simplify the arithmetic:

3x=6+9

Simplify the arithmetic:

3x=15

Divide both sides by :

(3x)3=153

Simplify the fraction:

x=153

Find the greatest common factor of the numerator and denominator:

x=(5·3)(1·3)

Factor out and cancel the greatest common factor:

x=5

11 additional steps

(x-9)=-(-(2x-6))

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

(x-9)=2x-6

Subtract from both sides:

(x-9)-2x=(2x-6)-2x

Group like terms:

(x-2x)-9=(2x-6)-2x

Simplify the arithmetic:

-x-9=(2x-6)-2x

Group like terms:

-x-9=(2x-2x)-6

Simplify the arithmetic:

x9=6

Add to both sides:

(-x-9)+9=-6+9

Simplify the arithmetic:

x=6+9

Simplify the arithmetic:

x=3

Multiply both sides by :

-x·-1=3·-1

Remove the one(s):

x=3·-1

Simplify the arithmetic:

x=3

4. List the solutions

x=5,3
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
y=|x9|
y=|2x6|
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.