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Solution - Absolute value equations

Exact form:

b = \frac{18}{5}, \frac{11}{5}

b=\frac{18}{5} , \frac{11}{5}

Mixed number form:

b = 3 \frac{3}{5}, 2 \frac{1}{5}

b=3\frac{3}{5} , 2\frac{1}{5}

Decimal form:

b = 3.6, 2.2

b=3.6 , 2.2

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
$| b - \frac{4}{5} | = | 3 b - 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| b - \frac{4}{5} \| = \| 3 b - 8 \|$
$x = + y$	$(b - \frac{4}{5}) = (3 b - 8)$
$x = - y$	$(b - \frac{4}{5}) = - (3 b - 8)$
$+ x = y$	$(b - \frac{4}{5}) = (3 b - 8)$
$- x = y$	$- (b - \frac{4}{5}) = (3 b - 8)$

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:

$\| x \| = \| y \|$	$\| b - \frac{4}{5} \| = \| 3 b - 8 \|$
$x = + y$ , $+ x = y$	$(b - \frac{4}{5}) = (3 b - 8)$
$x = - y$ , $- x = y$	$(b - \frac{4}{5}) = - (3 b - 8)$

2. Solve the two equations for b

17 additional steps

$(b + \frac{- 4}{5}) = (3 b - 8)$

Subtract from both sides:

$(b + \frac{- 4}{5}) - 3 b = (3 b - 8) - 3 b$

Group like terms:

$(b - 3 b) + \frac{- 4}{5} = (3 b - 8) - 3 b$

Simplify the arithmetic:

$- 2 b + \frac{- 4}{5} = (3 b - 8) - 3 b$

Group like terms:

$- 2 b + \frac{- 4}{5} = (3 b - 3 b) - 8$

Simplify the arithmetic:

$- 2 b + \frac{- 4}{5} = - 8$

Add to both sides:

$(- 2 b + \frac{- 4}{5}) + \frac{4}{5} = - 8 + \frac{4}{5}$

Combine the fractions:

$- 2 b + \frac{(- 4 + 4)}{5} = - 8 + \frac{4}{5}$

Combine the numerators:

$- 2 b + \frac{0}{5} = - 8 + \frac{4}{5}$

Reduce the zero numerator:

$- 2 b + 0 = - 8 + \frac{4}{5}$

Simplify the arithmetic:

$- 2 b = - 8 + \frac{4}{5}$

Convert the integer into a fraction:

$- 2 b = \frac{- 40}{5} + \frac{4}{5}$

Combine the fractions:

$- 2 b = \frac{(- 40 + 4)}{5}$

Combine the numerators:

$- 2 b = \frac{- 36}{5}$

Divide both sides by :

$\frac{(- 2 b)}{- 2} = \frac{(\frac{- 36}{5})}{- 2}$

Cancel out the negatives:

$\frac{2 b}{2} = \frac{(\frac{- 36}{5})}{- 2}$

Simplify the fraction:

$b = \frac{(\frac{- 36}{5})}{- 2}$

Simplify the arithmetic:

$b = \frac{- 36}{(5 \cdot - 2)}$

$b = \frac{18}{5}$

17 additional steps

$(b + \frac{- 4}{5}) = - (3 b - 8)$

Expand the parentheses:

$(b + \frac{- 4}{5}) = - 3 b + 8$

Add to both sides:

$(b + \frac{- 4}{5}) + 3 b = (- 3 b + 8) + 3 b$

Group like terms:

$(b + 3 b) + \frac{- 4}{5} = (- 3 b + 8) + 3 b$

Simplify the arithmetic:

$4 b + \frac{- 4}{5} = (- 3 b + 8) + 3 b$

Group like terms:

$4 b + \frac{- 4}{5} = (- 3 b + 3 b) + 8$

Simplify the arithmetic:

$4 b + \frac{- 4}{5} = 8$

Add to both sides:

$(4 b + \frac{- 4}{5}) + \frac{4}{5} = 8 + \frac{4}{5}$

Combine the fractions:

$4 b + \frac{(- 4 + 4)}{5} = 8 + \frac{4}{5}$

Combine the numerators:

$4 b + \frac{0}{5} = 8 + \frac{4}{5}$

Reduce the zero numerator:

$4 b + 0 = 8 + \frac{4}{5}$

Simplify the arithmetic:

$4 b = 8 + \frac{4}{5}$

Convert the integer into a fraction:

$4 b = \frac{40}{5} + \frac{4}{5}$

Combine the fractions:

$4 b = \frac{(40 + 4)}{5}$

Combine the numerators:

$4 b = \frac{44}{5}$

Divide both sides by :

$\frac{(4 b)}{4} = \frac{(\frac{44}{5})}{4}$

Simplify the fraction:

$b = \frac{(\frac{44}{5})}{4}$

Simplify the arithmetic:

$b = \frac{44}{(5 \cdot 4)}$

$b = \frac{11}{5}$

3. List the solutions

$b = \frac{18}{5}, \frac{11}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | b - \frac{4}{5} |$
$y = | 3 b - 8 |$
The equation is true where the two lines cross.

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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

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