Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

b = - \frac{9}{5}, - \frac{9}{7}

b=-\frac{9}{5} , -\frac{9}{7}

Mixed number form:

b = - 1 \frac{4}{5}, - 1 \frac{2}{7}

b=-1\frac{4}{5} , -1\frac{2}{7}

Decimal form:

b = - 1.8, - 1.286

b=-1.8 , -1.286

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| b | = | 6 b + 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| b \| = \| 6 b + 9 \|$
$x = + y$	$(b) = (6 b + 9)$
$x = - y$	$(b) = - (6 b + 9)$
$+ x = y$	$(b) = (6 b + 9)$
$- x = y$	$- (b) = (6 b + 9)$

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:

$\| x \| = \| y \|$	$\| b \| = \| 6 b + 9 \|$
$x = + y$ , $+ x = y$	$(b) = (6 b + 9)$
$x = - y$ , $- x = y$	$(b) = - (6 b + 9)$

2. Solve the two equations for b

7 additional steps

$b = (6 b + 9)$

Subtract from both sides:

$b - 6 b = (6 b + 9) - 6 b$

Simplify the arithmetic:

$- 5 b = (6 b + 9) - 6 b$

Group like terms:

$- 5 b = (6 b - 6 b) + 9$

Simplify the arithmetic:

$- 5 b = 9$

Divide both sides by :

$\frac{(- 5 b)}{- 5} = \frac{9}{- 5}$

Cancel out the negatives:

$\frac{5 b}{5} = \frac{9}{- 5}$

Simplify the fraction:

$b = \frac{9}{- 5}$

Move the negative sign from the denominator to the numerator:

$b = \frac{- 9}{5}$

6 additional steps

$b = - (6 b + 9)$

Expand the parentheses:

$b = - 6 b - 9$

Add to both sides:

$b + 6 b = (- 6 b - 9) + 6 b$

Simplify the arithmetic:

$7 b = (- 6 b - 9) + 6 b$

Group like terms:

$7 b = (- 6 b + 6 b) - 9$

Simplify the arithmetic:

$7 b = - 9$

Divide both sides by :

$\frac{(7 b)}{7} = \frac{- 9}{7}$

Simplify the fraction:

$b = \frac{- 9}{7}$

3. List the solutions

$b = - \frac{9}{5}, - \frac{9}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | b |$
$y = | 6 b + 9 |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

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.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for b

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for b

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved