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Solution - Absolute value equations

Exact form: x=-25,-89
x=-\frac{2}{5} , -\frac{8}{9}
Decimal form: x=0.4,0.889
x=-0.4 , -0.889

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
|7x+5|=|2x+3|
without the absolute value bars:

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

2. Solve the two equations for x

9 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Divide both sides by :

(5x)5=-25

Simplify the fraction:

x=-25

10 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

9x+5=(-2x-3)+2x

Group like terms:

9x+5=(-2x+2x)-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=8

Divide both sides by :

(9x)9=-89

Simplify the fraction:

x=-89

3. List the solutions

x=-25,-89
(2 solution(s))

4. Graph

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