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Solution - Absolute value equations

Exact form: x=-6,45
x=-6 , \frac{4}{5}
Decimal form: x=6,0.8
x=-6 , 0.8

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
|2x5|=|3x+1|
without the absolute value bars:

|x|=|y||2x5|=|3x+1|
x=+y(2x5)=(3x+1)
x=y(2x5)=(3x+1)
+x=y(2x5)=(3x+1)
x=y(2x5)=(3x+1)

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||2x5|=|3x+1|
x=+y , +x=y(2x5)=(3x+1)
x=y , x=y(2x5)=(3x+1)

2. Solve the two equations for x

10 additional steps

(2x-5)=(3x+1)

Subtract from both sides:

(2x-5)-3x=(3x+1)-3x

Group like terms:

(2x-3x)-5=(3x+1)-3x

Simplify the arithmetic:

-x-5=(3x+1)-3x

Group like terms:

-x-5=(3x-3x)+1

Simplify the arithmetic:

x5=1

Add to both sides:

(-x-5)+5=1+5

Simplify the arithmetic:

x=1+5

Simplify the arithmetic:

x=6

Multiply both sides by :

-x·-1=6·-1

Remove the one(s):

x=6·-1

Simplify the arithmetic:

x=6

10 additional steps

(2x-5)=-(3x+1)

Expand the parentheses:

(2x-5)=-3x-1

Add to both sides:

(2x-5)+3x=(-3x-1)+3x

Group like terms:

(2x+3x)-5=(-3x-1)+3x

Simplify the arithmetic:

5x-5=(-3x-1)+3x

Group like terms:

5x-5=(-3x+3x)-1

Simplify the arithmetic:

5x5=1

Add to both sides:

(5x-5)+5=-1+5

Simplify the arithmetic:

5x=1+5

Simplify the arithmetic:

5x=4

Divide both sides by :

(5x)5=45

Simplify the fraction:

x=45

3. List the solutions

x=-6,45
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|2x5|
y=|3x+1|
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.