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Solution - Absolute value equations

Exact form: y=65,-8
y=\frac{6}{5} , -8
Mixed number form: y=115,-8
y=1\frac{1}{5} , -8
Decimal form: y=1.2,8
y=1.2 , -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
|2y+7|=|3y+1|
without the absolute value bars:

|x|=|y||2y+7|=|3y+1|
x=+y(2y+7)=(3y+1)
x=y(2y+7)=(3y+1)
+x=y(2y+7)=(3y+1)
x=y(2y+7)=(3y+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||2y+7|=|3y+1|
x=+y , +x=y(2y+7)=(3y+1)
x=y , x=y(2y+7)=(3y+1)

2. Solve the two equations for y

11 additional steps

(-2y+7)=(3y+1)

Subtract from both sides:

(-2y+7)-3y=(3y+1)-3y

Group like terms:

(-2y-3y)+7=(3y+1)-3y

Simplify the arithmetic:

-5y+7=(3y+1)-3y

Group like terms:

-5y+7=(3y-3y)+1

Simplify the arithmetic:

5y+7=1

Subtract from both sides:

(-5y+7)-7=1-7

Simplify the arithmetic:

5y=17

Simplify the arithmetic:

5y=6

Divide both sides by :

(-5y)-5=-6-5

Cancel out the negatives:

5y5=-6-5

Simplify the fraction:

y=-6-5

Cancel out the negatives:

y=65

8 additional steps

(-2y+7)=-(3y+1)

Expand the parentheses:

(-2y+7)=-3y-1

Add to both sides:

(-2y+7)+3y=(-3y-1)+3y

Group like terms:

(-2y+3y)+7=(-3y-1)+3y

Simplify the arithmetic:

y+7=(-3y-1)+3y

Group like terms:

y+7=(-3y+3y)-1

Simplify the arithmetic:

y+7=1

Subtract from both sides:

(y+7)-7=-1-7

Simplify the arithmetic:

y=17

Simplify the arithmetic:

y=8

3. List the solutions

y=65,-8
(2 solution(s))

4. Graph

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