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Solution - Absolute value equations

Exact form: x=174,1714
x=\frac{17}{4} , \frac{17}{14}
Mixed number form: x=414,1314
x=4\frac{1}{4} , 1\frac{3}{14}
Decimal form: x=4.25,1.214
x=4.25 , 1.214

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
|9x17|=|5x|
without the absolute value bars:

|x|=|y||9x17|=|5x|
x=+y(9x17)=(5x)
x=y(9x17)=(5x)
+x=y(9x17)=(5x)
x=y(9x17)=(5x)

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||9x17|=|5x|
x=+y , +x=y(9x17)=(5x)
x=y , x=y(9x17)=(5x)

2. Solve the two equations for x

8 additional steps

(9x-17)=5x

Subtract from both sides:

(9x-17)-5x=(5x)-5x

Group like terms:

(9x-5x)-17=(5x)-5x

Simplify the arithmetic:

4x-17=(5x)-5x

Simplify the arithmetic:

4x17=0

Add to both sides:

(4x-17)+17=0+17

Simplify the arithmetic:

4x=0+17

Simplify the arithmetic:

4x=17

Divide both sides by :

(4x)4=174

Simplify the fraction:

x=174

7 additional steps

(9x-17)=-5x

Add to both sides:

(9x-17)+17=(-5x)+17

Simplify the arithmetic:

9x=(-5x)+17

Add to both sides:

(9x)+5x=((-5x)+17)+5x

Simplify the arithmetic:

14x=((-5x)+17)+5x

Group like terms:

14x=(-5x+5x)+17

Simplify the arithmetic:

14x=17

Divide both sides by :

(14x)14=1714

Simplify the fraction:

x=1714

3. List the solutions

x=174,1714
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|9x17|
y=|5x|
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.