Enter an equation or problem
Camera input is not recognized!

Solution - Absolute value equations

Exact form: y=-94
y=-\frac{9}{4}
Mixed number form: y=-214
y=-2\frac{1}{4}
Decimal form: y=2.25
y=-2.25

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+3|=|2y+6|
without the absolute value bars:

|x|=|y||2y+3|=|2y+6|
x=+y(2y+3)=(2y+6)
x=y(2y+3)=(2y+6)
+x=y(2y+3)=(2y+6)
x=y(2y+3)=(2y+6)

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+3|=|2y+6|
x=+y , +x=y(2y+3)=(2y+6)
x=y , x=y(2y+3)=(2y+6)

2. Solve the two equations for y

5 additional steps

(2y+3)=(2y+6)

Subtract from both sides:

(2y+3)-2y=(2y+6)-2y

Group like terms:

(2y-2y)+3=(2y+6)-2y

Simplify the arithmetic:

3=(2y+6)-2y

Group like terms:

3=(2y-2y)+6

Simplify the arithmetic:

3=6

The statement is false:

3=6

The equation is false so it has no solution.

10 additional steps

(2y+3)=-(2y+6)

Expand the parentheses:

(2y+3)=-2y-6

Add to both sides:

(2y+3)+2y=(-2y-6)+2y

Group like terms:

(2y+2y)+3=(-2y-6)+2y

Simplify the arithmetic:

4y+3=(-2y-6)+2y

Group like terms:

4y+3=(-2y+2y)-6

Simplify the arithmetic:

4y+3=6

Subtract from both sides:

(4y+3)-3=-6-3

Simplify the arithmetic:

4y=63

Simplify the arithmetic:

4y=9

Divide both sides by :

(4y)4=-94

Simplify the fraction:

y=-94

3. Graph

Each line represents the function of one side of the equation:
y=|2y+3|
y=|2y+6|
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.