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Solution - Absolute value equations

Exact form: y=8,4
y=8 , -4

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
|4y+4|=|2y+20|
without the absolute value bars:

|x|=|y||4y+4|=|2y+20|
x=+y(4y+4)=(2y+20)
x=y(4y+4)=(2y+20)
+x=y(4y+4)=(2y+20)
x=y(4y+4)=(2y+20)

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||4y+4|=|2y+20|
x=+y , +x=y(4y+4)=(2y+20)
x=y , x=y(4y+4)=(2y+20)

2. Solve the two equations for y

11 additional steps

(4y+4)=(2y+20)

Subtract from both sides:

(4y+4)-2y=(2y+20)-2y

Group like terms:

(4y-2y)+4=(2y+20)-2y

Simplify the arithmetic:

2y+4=(2y+20)-2y

Group like terms:

2y+4=(2y-2y)+20

Simplify the arithmetic:

2y+4=20

Subtract from both sides:

(2y+4)-4=20-4

Simplify the arithmetic:

2y=204

Simplify the arithmetic:

2y=16

Divide both sides by :

(2y)2=162

Simplify the fraction:

y=162

Find the greatest common factor of the numerator and denominator:

y=(8·2)(1·2)

Factor out and cancel the greatest common factor:

y=8

12 additional steps

(4y+4)=-(2y+20)

Expand the parentheses:

(4y+4)=-2y-20

Add to both sides:

(4y+4)+2y=(-2y-20)+2y

Group like terms:

(4y+2y)+4=(-2y-20)+2y

Simplify the arithmetic:

6y+4=(-2y-20)+2y

Group like terms:

6y+4=(-2y+2y)-20

Simplify the arithmetic:

6y+4=20

Subtract from both sides:

(6y+4)-4=-20-4

Simplify the arithmetic:

6y=204

Simplify the arithmetic:

6y=24

Divide both sides by :

(6y)6=-246

Simplify the fraction:

y=-246

Find the greatest common factor of the numerator and denominator:

y=(-4·6)(1·6)

Factor out and cancel the greatest common factor:

y=4

3. List the solutions

y=8,4
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|4y+4|
y=|2y+20|
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.