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Solution - Absolute value equations

Exact form: b=12,1
b=\frac{1}{2} , 1
Decimal form: b=0.5,1
b=0.5 , 1

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
|b|=|3b2|
without the absolute value bars:

|x|=|y||b|=|3b2|
x=+y(b)=(3b2)
x=y(b)=(3b2)
+x=y(b)=(3b2)
x=y((b))=(3b2)

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||b|=|3b2|
x=+y , +x=y(b)=(3b2)
x=y , x=y(b)=(3b2)

2. Solve the two equations for b

9 additional steps

-b=(3b-2)

Subtract from both sides:

-b-3b=(3b-2)-3b

Simplify the arithmetic:

-4b=(3b-2)-3b

Group like terms:

-4b=(3b-3b)-2

Simplify the arithmetic:

-4b=-2

Divide both sides by :

(-4b)-4=-2-4

Cancel out the negatives:

4b4=-2-4

Simplify the fraction:

b=-2-4

Cancel out the negatives:

b=24

Find the greatest common factor of the numerator and denominator:

b=(1·2)(2·2)

Factor out and cancel the greatest common factor:

b=12

7 additional steps

-b=-(3b-2)

Expand the parentheses:

-b=-3b+2

Add to both sides:

-b+3b=(-3b+2)+3b

Simplify the arithmetic:

2b=(-3b+2)+3b

Group like terms:

2b=(-3b+3b)+2

Simplify the arithmetic:

2b=2

Divide both sides by :

(2b)2=22

Simplify the fraction:

b=22

Simplify the fraction:

b=1

3. List the solutions

b=12,1
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|b|
y=|3b2|
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.