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Solution - Absolute value equations

Exact form: x=0,2
x=0 , 2

Other Ways to Solve

Absolute value equations

Step-by-step explanation

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

|2x+3|+|5x3|=0

Add |5x3| to both sides of the equation:

|2x+3|+|5x3||5x3|=|5x3|

Simplify the arithmetic

|2x+3|=|5x3|

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
|2x+3|=|5x3|
without the absolute value bars:

|x|=|y||2x+3|=|5x3|
x=+y(2x+3)=(5x3)
x=y(2x+3)=(5x3)
+x=y(2x+3)=(5x3)
x=y(2x+3)=(5x3)

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+3|=|5x3|
x=+y , +x=y(2x+3)=(5x3)
x=y , x=y(2x+3)=(5x3)

3. Solve the two equations for x

9 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

7x+3=(-5x+3)+5x

Group like terms:

7x+3=(-5x+5x)+3

Simplify the arithmetic:

7x+3=3

Subtract from both sides:

(7x+3)-3=3-3

Simplify the arithmetic:

7x=33

Simplify the arithmetic:

7x=0

Divide both sides by the coefficient:

x=0

14 additional steps

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

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

(2x+3)=5x-3

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

3x+3=3

Subtract from both sides:

(-3x+3)-3=-3-3

Simplify the arithmetic:

3x=33

Simplify the arithmetic:

3x=6

Divide both sides by :

(-3x)-3=-6-3

Cancel out the negatives:

3x3=-6-3

Simplify the fraction:

x=-6-3

Cancel out the negatives:

x=63

Find the greatest common factor of the numerator and denominator:

x=(2·3)(1·3)

Factor out and cancel the greatest common factor:

x=2

4. List the solutions

x=0,2
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
y=|2x+3|
y=|5x3|
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.