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Solution - Absolute value equations

Exact form: x=278,2710
x=\frac{27}{8} , \frac{27}{10}
Mixed number form: x=338,2710
x=3\frac{3}{8} , 2\frac{7}{10}
Decimal form: x=3.375,2.7
x=3.375 , 2.7

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|=9|x3|
without the absolute value bars:

|x|=|y||x|=9|x3|
x=+y(x)=9(x3)
x=y(x)=9((x3))
+x=y(x)=9(x3)
x=y(x)=9(x3)

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|=9|x3|
x=+y , +x=y(x)=9(x3)
x=y , x=y(x)=9((x3))

2. Solve the two equations for x

9 additional steps

x=9·(x-3)

Expand the parentheses:

x=9x+9·-3

Simplify the arithmetic:

x=9x27

Subtract from both sides:

x-9x=(9x-27)-9x

Simplify the arithmetic:

-8x=(9x-27)-9x

Group like terms:

-8x=(9x-9x)-27

Simplify the arithmetic:

8x=27

Divide both sides by :

(-8x)-8=-27-8

Cancel out the negatives:

8x8=-27-8

Simplify the fraction:

x=-27-8

Cancel out the negatives:

x=278

10 additional steps

x=9·(-(x-3))

Expand the parentheses:

x=9·(-x+3)

x=9·-x+9·3

Group like terms:

x=(9·-1)x+9·3

Multiply the coefficients:

x=-9x+9·3

Simplify the arithmetic:

x=9x+27

Add to both sides:

x+9x=(-9x+27)+9x

Simplify the arithmetic:

10x=(-9x+27)+9x

Group like terms:

10x=(-9x+9x)+27

Simplify the arithmetic:

10x=27

Divide both sides by :

(10x)10=2710

Simplify the fraction:

x=2710

3. List the solutions

x=278,2710
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|x|
y=9|x3|
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.