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Solution - Absolute value equations

Exact form: x=85,-29
x=\frac{8}{5} , -\frac{2}{9}
Mixed number form: x=135,-29
x=1\frac{3}{5} , -\frac{2}{9}
Decimal form: x=1.6,0.222
x=1.6 , -0.222

Other Ways to Solve

Absolute value equations

Step-by-step explanation

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

|2x+5|+|7x+3|=0

Add |7x+3| to both sides of the equation:

|2x+5|+|7x+3||7x+3|=|7x+3|

Simplify the arithmetic

|2x+5|=|7x+3|

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+5|=|7x+3|
without the absolute value bars:

|x|=|y||2x+5|=|7x+3|
x=+y(2x+5)=(7x+3)
x=y(2x+5)=(7x+3)
+x=y(2x+5)=(7x+3)
x=y(2x+5)=(7x+3)

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|=|7x+3|
x=+y , +x=y(2x+5)=(7x+3)
x=y , x=y(2x+5)=(7x+3)

3. Solve the two equations for x

12 additional steps

(2x+5)=-(-7x+3)

Expand the parentheses:

(2x+5)=7x-3

Subtract from both sides:

(2x+5)-7x=(7x-3)-7x

Group like terms:

(2x-7x)+5=(7x-3)-7x

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

5x+5=3

Subtract from both sides:

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

Simplify the arithmetic:

5x=35

Simplify the arithmetic:

5x=8

Divide both sides by :

(-5x)-5=-8-5

Cancel out the negatives:

5x5=-8-5

Simplify the fraction:

x=-8-5

Cancel out the negatives:

x=85

10 additional steps

(2x+5)=-(-(-7x+3))

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

(2x+5)=-7x+3

Add to both sides:

(2x+5)+7x=(-7x+3)+7x

Group like terms:

(2x+7x)+5=(-7x+3)+7x

Simplify the arithmetic:

9x+5=(-7x+3)+7x

Group like terms:

9x+5=(-7x+7x)+3

Simplify the arithmetic:

9x+5=3

Subtract from both sides:

(9x+5)-5=3-5

Simplify the arithmetic:

9x=35

Simplify the arithmetic:

9x=2

Divide both sides by :

(9x)9=-29

Simplify the fraction:

x=-29

4. List the solutions

x=85,-29
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
y=|2x+5|
y=|7x+3|
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.