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Solution - Absolute value equations

Exact form: x=9,1
x=9 , 1

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
|5x9|=|4x|
without the absolute value bars:

|x|=|y||5x9|=|4x|
x=+y(5x9)=(4x)
x=y(5x9)=(4x)
+x=y(5x9)=(4x)
x=y(5x9)=(4x)

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||5x9|=|4x|
x=+y , +x=y(5x9)=(4x)
x=y , x=y(5x9)=(4x)

2. Solve the two equations for x

6 additional steps

(5x-9)=4x

Subtract from both sides:

(5x-9)-4x=(4x)-4x

Group like terms:

(5x-4x)-9=(4x)-4x

Simplify the arithmetic:

x-9=(4x)-4x

Simplify the arithmetic:

x9=0

Add to both sides:

(x-9)+9=0+9

Simplify the arithmetic:

x=0+9

Simplify the arithmetic:

x=9

8 additional steps

(5x-9)=-4x

Add to both sides:

(5x-9)+9=(-4x)+9

Simplify the arithmetic:

5x=(-4x)+9

Add to both sides:

(5x)+4x=((-4x)+9)+4x

Simplify the arithmetic:

9x=((-4x)+9)+4x

Group like terms:

9x=(-4x+4x)+9

Simplify the arithmetic:

9x=9

Divide both sides by :

(9x)9=99

Simplify the fraction:

x=99

Simplify the fraction:

x=1

3. List the solutions

x=9,1
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|5x9|
y=|4x|
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.