Camera input is not recognized!

Solution - Absolute value equations

Exact form: x=56,-54
x=\frac{5}{6} , -\frac{5}{4}
Mixed number form: x=56,-114
x=\frac{5}{6} , -1\frac{1}{4}
Decimal form: x=0.833,1.25
x=0.833 , -1.25

Step-by-step explanation

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

|5x||x+5|=0

Add |x+5| to both sides of the equation:

|5x||x+5|+|x+5|=|x+5|

Simplify the arithmetic

|5x|=|x+5|

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
|5x|=|x+5|
without the absolute value bars:

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

3. Solve the two equations for x

5 additional steps

5x=(-x+5)

Add to both sides:

(5x)+x=(-x+5)+x

Simplify the arithmetic:

6x=(-x+5)+x

Group like terms:

6x=(-x+x)+5

Simplify the arithmetic:

6x=5

Divide both sides by :

(6x)6=56

Simplify the fraction:

x=56

6 additional steps

5x=-(-x+5)

Expand the parentheses:

5x=x5

Subtract from both sides:

(5x)-x=(x-5)-x

Simplify the arithmetic:

4x=(x-5)-x

Group like terms:

4x=(x-x)-5

Simplify the arithmetic:

4x=5

Divide both sides by :

(4x)4=-54

Simplify the fraction:

x=-54

4. List the solutions

x=56,-54
(2 solution(s))

5. Graph

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