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Solution - Absolute value equations

Exact form:

x = \frac{44}{3}, - \frac{56}{13}

x=\frac{44}{3} , -\frac{56}{13}

Mixed number form:

x = 14 \frac{2}{3}, - 4 \frac{4}{13}

x=14\frac{2}{3} , -4\frac{4}{13}

Decimal form:

x = 14.667, - 4.308

x=14.667 , -4.308

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
$| \frac{4}{5} x + \frac{3}{5} | = | \frac{1}{2} x + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| \frac{4}{5} x + \frac{3}{5} \| = \| \frac{1}{2} x + 5 \|$
$x = + y$	$(\frac{4}{5} x + \frac{3}{5}) = (\frac{1}{2} x + 5)$
$x = - y$	$(\frac{4}{5} x + \frac{3}{5}) = - (\frac{1}{2} x + 5)$
$+ x = y$	$(\frac{4}{5} x + \frac{3}{5}) = (\frac{1}{2} x + 5)$
$- x = y$	$- (\frac{4}{5} x + \frac{3}{5}) = (\frac{1}{2} x + 5)$

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:

$\| x \| = \| y \|$	$\| \frac{4}{5} x + \frac{3}{5} \| = \| \frac{1}{2} x + 5 \|$
$x = + y$ , $+ x = y$	$(\frac{4}{5} x + \frac{3}{5}) = (\frac{1}{2} x + 5)$
$x = - y$ , $- x = y$	$(\frac{4}{5} x + \frac{3}{5}) = - (\frac{1}{2} x + 5)$

2. Solve the two equations for x

26 additional steps

$(\frac{4}{5} \cdot x + \frac{3}{5}) = (\frac{1}{2} x + 5)$

Subtract from both sides:

$(\frac{4}{5} x + \frac{3}{5}) - \frac{1}{2} \cdot x = (\frac{1}{2} x + 5) - \frac{1}{2} x$

Group like terms:

$(\frac{4}{5} \cdot x + \frac{- 1}{2} \cdot x) + \frac{3}{5} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Group the coefficients:

$(\frac{4}{5} + \frac{- 1}{2}) x + \frac{3}{5} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Find the lowest common denominator:

$(\frac{(4 \cdot 2)}{(5 \cdot 2)} + \frac{(- 1 \cdot 5)}{(2 \cdot 5)}) x + \frac{3}{5} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Multiply the denominators:

$(\frac{(4 \cdot 2)}{10} + \frac{(- 1 \cdot 5)}{10}) x + \frac{3}{5} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Multiply the numerators:

$(\frac{8}{10} + \frac{- 5}{10}) x + \frac{3}{5} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Combine the fractions:

$\frac{(8 - 5)}{10} \cdot x + \frac{3}{5} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Combine the numerators:

$\frac{3}{10} \cdot x + \frac{3}{5} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Group like terms:

$\frac{3}{10} \cdot x + \frac{3}{5} = (\frac{1}{2} \cdot x + \frac{- 1}{2} x) + 5$

Combine the fractions:

$\frac{3}{10} \cdot x + \frac{3}{5} = \frac{(1 - 1)}{2} x + 5$

Combine the numerators:

$\frac{3}{10} \cdot x + \frac{3}{5} = \frac{0}{2} x + 5$

Reduce the zero numerator:

$\frac{3}{10} x + \frac{3}{5} = 0 x + 5$

Simplify the arithmetic:

$\frac{3}{10} x + \frac{3}{5} = 5$

Subtract from both sides:

$(\frac{3}{10} x + \frac{3}{5}) - \frac{3}{5} = 5 - \frac{3}{5}$

Combine the fractions:

$\frac{3}{10} x + \frac{(3 - 3)}{5} = 5 - \frac{3}{5}$

Combine the numerators:

$\frac{3}{10} x + \frac{0}{5} = 5 - \frac{3}{5}$

Reduce the zero numerator:

$\frac{3}{10} x + 0 = 5 - \frac{3}{5}$

Simplify the arithmetic:

$\frac{3}{10} x = 5 - \frac{3}{5}$

Convert the integer into a fraction:

$\frac{3}{10} x = \frac{25}{5} + \frac{- 3}{5}$

Combine the fractions:

$\frac{3}{10} x = \frac{(25 - 3)}{5}$

Combine the numerators:

$\frac{3}{10} x = \frac{22}{5}$

Multiply both sides by inverse fraction :

$(\frac{3}{10} x) \cdot \frac{10}{3} = (\frac{22}{5}) \cdot \frac{10}{3}$

Group like terms:

$(\frac{3}{10} \cdot \frac{10}{3}) x = (\frac{22}{5}) \cdot \frac{10}{3}$

Multiply the coefficients:

$\frac{(3 \cdot 10)}{(10 \cdot 3)} x = (\frac{22}{5}) \cdot \frac{10}{3}$

Simplify the fraction:

$x = (\frac{22}{5}) \cdot \frac{10}{3}$

Multiply the fraction(s):

$x = \frac{(22 \cdot 10)}{(5 \cdot 3)}$

Simplify the arithmetic:

$x = \frac{44}{3}$

27 additional steps

$(\frac{4}{5} x + \frac{3}{5}) = - (\frac{1}{2} x + 5)$

Expand the parentheses:

$(\frac{4}{5} \cdot x + \frac{3}{5}) = \frac{- 1}{2} x - 5$

Add to both sides:

$(\frac{4}{5} x + \frac{3}{5}) + \frac{1}{2} \cdot x = (\frac{- 1}{2} x - 5) + \frac{1}{2} x$

Group like terms:

$(\frac{4}{5} \cdot x + \frac{1}{2} \cdot x) + \frac{3}{5} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Group the coefficients:

$(\frac{4}{5} + \frac{1}{2}) x + \frac{3}{5} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Find the lowest common denominator:

$(\frac{(4 \cdot 2)}{(5 \cdot 2)} + \frac{(1 \cdot 5)}{(2 \cdot 5)}) x + \frac{3}{5} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Multiply the denominators:

$(\frac{(4 \cdot 2)}{10} + \frac{(1 \cdot 5)}{10}) x + \frac{3}{5} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Multiply the numerators:

$(\frac{8}{10} + \frac{5}{10}) x + \frac{3}{5} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Combine the fractions:

$\frac{(8 + 5)}{10} \cdot x + \frac{3}{5} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Combine the numerators:

$\frac{13}{10} \cdot x + \frac{3}{5} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Group like terms:

$\frac{13}{10} \cdot x + \frac{3}{5} = (\frac{- 1}{2} \cdot x + \frac{1}{2} x) - 5$

Combine the fractions:

$\frac{13}{10} \cdot x + \frac{3}{5} = \frac{(- 1 + 1)}{2} x - 5$

Combine the numerators:

$\frac{13}{10} \cdot x + \frac{3}{5} = \frac{0}{2} x - 5$

Reduce the zero numerator:

$\frac{13}{10} x + \frac{3}{5} = 0 x - 5$

Simplify the arithmetic:

$\frac{13}{10} x + \frac{3}{5} = - 5$

Subtract from both sides:

$(\frac{13}{10} x + \frac{3}{5}) - \frac{3}{5} = - 5 - \frac{3}{5}$

Combine the fractions:

$\frac{13}{10} x + \frac{(3 - 3)}{5} = - 5 - \frac{3}{5}$

Combine the numerators:

$\frac{13}{10} x + \frac{0}{5} = - 5 - \frac{3}{5}$

Reduce the zero numerator:

$\frac{13}{10} x + 0 = - 5 - \frac{3}{5}$

Simplify the arithmetic:

$\frac{13}{10} x = - 5 - \frac{3}{5}$

Convert the integer into a fraction:

$\frac{13}{10} x = \frac{- 25}{5} + \frac{- 3}{5}$

Combine the fractions:

$\frac{13}{10} x = \frac{(- 25 - 3)}{5}$

Combine the numerators:

$\frac{13}{10} x = \frac{- 28}{5}$

Multiply both sides by inverse fraction :

$(\frac{13}{10} x) \cdot \frac{10}{13} = (\frac{- 28}{5}) \cdot \frac{10}{13}$

Group like terms:

$(\frac{13}{10} \cdot \frac{10}{13}) x = (\frac{- 28}{5}) \cdot \frac{10}{13}$

Multiply the coefficients:

$\frac{(13 \cdot 10)}{(10 \cdot 13)} x = (\frac{- 28}{5}) \cdot \frac{10}{13}$

Simplify the fraction:

$x = (\frac{- 28}{5}) \cdot \frac{10}{13}$

Multiply the fraction(s):

$x = \frac{(- 28 \cdot 10)}{(5 \cdot 13)}$

Simplify the arithmetic:

$x = \frac{- 56}{13}$

3. List the solutions

$x = \frac{44}{3}, - \frac{56}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{4}{5} x + \frac{3}{5} |$
$y = | \frac{1}{2} x + 5 |$
The equation is true where the two lines cross.

How did we do?

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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 x

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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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