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Solution - Absolute value equations

Exact form: y=3
y=3

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
|3y7|=|3y11|
without the absolute value bars:

|x|=|y||3y7|=|3y11|
x=+y(3y7)=(3y11)
x=y(3y7)=(3y11)
+x=y(3y7)=(3y11)
x=y(3y7)=(3y11)

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||3y7|=|3y11|
x=+y , +x=y(3y7)=(3y11)
x=y , x=y(3y7)=(3y11)

2. Solve the two equations for y

5 additional steps

(3y-7)=(3y-11)

Subtract from both sides:

(3y-7)-3y=(3y-11)-3y

Group like terms:

(3y-3y)-7=(3y-11)-3y

Simplify the arithmetic:

-7=(3y-11)-3y

Group like terms:

-7=(3y-3y)-11

Simplify the arithmetic:

7=11

The statement is false:

7=11

The equation is false so it has no solution.

12 additional steps

(3y-7)=-(3y-11)

Expand the parentheses:

(3y-7)=-3y+11

Add to both sides:

(3y-7)+3y=(-3y+11)+3y

Group like terms:

(3y+3y)-7=(-3y+11)+3y

Simplify the arithmetic:

6y-7=(-3y+11)+3y

Group like terms:

6y-7=(-3y+3y)+11

Simplify the arithmetic:

6y7=11

Add to both sides:

(6y-7)+7=11+7

Simplify the arithmetic:

6y=11+7

Simplify the arithmetic:

6y=18

Divide both sides by :

(6y)6=186

Simplify the fraction:

y=186

Find the greatest common factor of the numerator and denominator:

y=(3·6)(1·6)

Factor out and cancel the greatest common factor:

y=3

3. Graph

Each line represents the function of one side of the equation:
y=|3y7|
y=|3y11|
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.