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Solution - Absolute value equations

Exact form: b=3,7
b=3 , -7

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
|3b4|=|2b+11|
without the absolute value bars:

|x|=|y||3b4|=|2b+11|
x=+y(3b4)=(2b+11)
x=y(3b4)=(2b+11)
+x=y(3b4)=(2b+11)
x=y(3b4)=(2b+11)

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||3b4|=|2b+11|
x=+y , +x=y(3b4)=(2b+11)
x=y , x=y(3b4)=(2b+11)

2. Solve the two equations for b

11 additional steps

(3b-4)=(-2b+11)

Add to both sides:

(3b-4)+2b=(-2b+11)+2b

Group like terms:

(3b+2b)-4=(-2b+11)+2b

Simplify the arithmetic:

5b-4=(-2b+11)+2b

Group like terms:

5b-4=(-2b+2b)+11

Simplify the arithmetic:

5b-4=11

Add to both sides:

(5b-4)+4=11+4

Simplify the arithmetic:

5b=11+4

Simplify the arithmetic:

5b=15

Divide both sides by :

(5b)5=155

Simplify the fraction:

b=155

Find the greatest common factor of the numerator and denominator:

b=(3·5)(1·5)

Factor out and cancel the greatest common factor:

b=3

8 additional steps

(3b-4)=-(-2b+11)

Expand the parentheses:

(3b-4)=2b-11

Subtract from both sides:

(3b-4)-2b=(2b-11)-2b

Group like terms:

(3b-2b)-4=(2b-11)-2b

Simplify the arithmetic:

b-4=(2b-11)-2b

Group like terms:

b-4=(2b-2b)-11

Simplify the arithmetic:

b-4=-11

Add to both sides:

(b-4)+4=-11+4

Simplify the arithmetic:

b=-11+4

Simplify the arithmetic:

b=-7

3. List the solutions

b=3,7
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|3b4|
y=|2b+11|
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.