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Solution - Absolute value equations

Exact form: x=53,5
x=\frac{5}{3} , 5
Mixed number form: x=123,5
x=1\frac{2}{3} , 5
Decimal form: x=1.667,5
x=1.667 , 5

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
|2x+5|=|x|
without the absolute value bars:

|x|=|y||2x+5|=|x|
x=+y(2x+5)=(x)
x=y(2x+5)=(x)
+x=y(2x+5)=(x)
x=y(2x+5)=(x)

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||2x+5|=|x|
x=+y , +x=y(2x+5)=(x)
x=y , x=y(2x+5)=(x)

2. Solve the two equations for x

10 additional steps

(-2x+5)=x

Subtract from both sides:

(-2x+5)-x=x-x

Group like terms:

(-2x-x)+5=x-x

Simplify the arithmetic:

3x+5=xx

Simplify the arithmetic:

3x+5=0

Subtract from both sides:

(-3x+5)-5=0-5

Simplify the arithmetic:

3x=05

Simplify the arithmetic:

3x=5

Divide both sides by :

(-3x)-3=-5-3

Cancel out the negatives:

3x3=-5-3

Simplify the fraction:

x=-5-3

Cancel out the negatives:

x=53

9 additional steps

(-2x+5)=-x

Add to both sides:

(-2x+5)+x=-x+x

Group like terms:

(-2x+x)+5=-x+x

Simplify the arithmetic:

x+5=x+x

Simplify the arithmetic:

x+5=0

Subtract from both sides:

(-x+5)-5=0-5

Simplify the arithmetic:

x=05

Simplify the arithmetic:

x=5

Multiply both sides by :

-x·-1=-5·-1

Remove the one(s):

x=-5·-1

Simplify the arithmetic:

x=5

3. List the solutions

x=53,5
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|2x+5|
y=|x|
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.