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Solution - Absolute value equations

Exact form:

x = \frac{100}{9}, - \frac{140}{33}

x=\frac{100}{9} , -\frac{140}{33}

Mixed number form:

x = 11 \frac{1}{9}, - 4 \frac{8}{33}

x=11\frac{1}{9} , -4\frac{8}{33}

Decimal form:

x = 11.111, - 4.242

x=11.111 , -4.242

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
$| \frac{7}{8} x + \frac{5}{6} | = | \frac{1}{2} x + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| \frac{7}{8} x + \frac{5}{6} \| = \| \frac{1}{2} x + 5 \|$
$x = + y$	$(\frac{7}{8} x + \frac{5}{6}) = (\frac{1}{2} x + 5)$
$x = - y$	$(\frac{7}{8} x + \frac{5}{6}) = - (\frac{1}{2} x + 5)$
$+ x = y$	$(\frac{7}{8} x + \frac{5}{6}) = (\frac{1}{2} x + 5)$
$- x = y$	$- (\frac{7}{8} x + \frac{5}{6}) = (\frac{1}{2} x + 5)$

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:

$\| x \| = \| y \|$	$\| \frac{7}{8} x + \frac{5}{6} \| = \| \frac{1}{2} x + 5 \|$
$x = + y$ , $+ x = y$	$(\frac{7}{8} x + \frac{5}{6}) = (\frac{1}{2} x + 5)$
$x = - y$ , $- x = y$	$(\frac{7}{8} x + \frac{5}{6}) = - (\frac{1}{2} x + 5)$

2. Solve the two equations for x

27 additional steps

$(\frac{7}{8} \cdot x + \frac{5}{6}) = (\frac{1}{2} x + 5)$

Subtract from both sides:

$(\frac{7}{8} x + \frac{5}{6}) - \frac{1}{2} \cdot x = (\frac{1}{2} x + 5) - \frac{1}{2} x$

Group like terms:

$(\frac{7}{8} \cdot x + \frac{- 1}{2} \cdot x) + \frac{5}{6} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Group the coefficients:

$(\frac{7}{8} + \frac{- 1}{2}) x + \frac{5}{6} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Find the lowest common denominator:

$(\frac{7}{8} + \frac{(- 1 \cdot 4)}{(2 \cdot 4)}) x + \frac{5}{6} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Multiply the denominators:

$(\frac{7}{8} + \frac{(- 1 \cdot 4)}{8}) x + \frac{5}{6} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Multiply the numerators:

$(\frac{7}{8} + \frac{- 4}{8}) x + \frac{5}{6} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Combine the fractions:

$\frac{(7 - 4)}{8} \cdot x + \frac{5}{6} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Combine the numerators:

$\frac{3}{8} \cdot x + \frac{5}{6} = (\frac{1}{2} \cdot x + 5) - \frac{1}{2} x$

Group like terms:

$\frac{3}{8} \cdot x + \frac{5}{6} = (\frac{1}{2} \cdot x + \frac{- 1}{2} x) + 5$

Combine the fractions:

$\frac{3}{8} \cdot x + \frac{5}{6} = \frac{(1 - 1)}{2} x + 5$

Combine the numerators:

$\frac{3}{8} \cdot x + \frac{5}{6} = \frac{0}{2} x + 5$

Reduce the zero numerator:

$\frac{3}{8} x + \frac{5}{6} = 0 x + 5$

Simplify the arithmetic:

$\frac{3}{8} x + \frac{5}{6} = 5$

Subtract from both sides:

$(\frac{3}{8} x + \frac{5}{6}) - \frac{5}{6} = 5 - \frac{5}{6}$

Combine the fractions:

$\frac{3}{8} x + \frac{(5 - 5)}{6} = 5 - \frac{5}{6}$

Combine the numerators:

$\frac{3}{8} x + \frac{0}{6} = 5 - \frac{5}{6}$

Reduce the zero numerator:

$\frac{3}{8} x + 0 = 5 - \frac{5}{6}$

Simplify the arithmetic:

$\frac{3}{8} x = 5 - \frac{5}{6}$

Convert the integer into a fraction:

$\frac{3}{8} x = \frac{30}{6} + \frac{- 5}{6}$

Combine the fractions:

$\frac{3}{8} x = \frac{(30 - 5)}{6}$

Combine the numerators:

$\frac{3}{8} x = \frac{25}{6}$

Multiply both sides by inverse fraction :

$(\frac{3}{8} x) \cdot \frac{8}{3} = (\frac{25}{6}) \cdot \frac{8}{3}$

Group like terms:

$(\frac{3}{8} \cdot \frac{8}{3}) x = (\frac{25}{6}) \cdot \frac{8}{3}$

Multiply the coefficients:

$\frac{(3 \cdot 8)}{(8 \cdot 3)} x = (\frac{25}{6}) \cdot \frac{8}{3}$

Simplify the fraction:

$x = (\frac{25}{6}) \cdot \frac{8}{3}$

Multiply the fraction(s):

$x = \frac{(25 \cdot 8)}{(6 \cdot 3)}$

Simplify the arithmetic:

$x = \frac{100}{(3 \cdot 3)}$

$x = \frac{100}{9}$

28 additional steps

$(\frac{7}{8} x + \frac{5}{6}) = - (\frac{1}{2} x + 5)$

Expand the parentheses:

$(\frac{7}{8} \cdot x + \frac{5}{6}) = \frac{- 1}{2} x - 5$

Add to both sides:

$(\frac{7}{8} x + \frac{5}{6}) + \frac{1}{2} \cdot x = (\frac{- 1}{2} x - 5) + \frac{1}{2} x$

Group like terms:

$(\frac{7}{8} \cdot x + \frac{1}{2} \cdot x) + \frac{5}{6} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Group the coefficients:

$(\frac{7}{8} + \frac{1}{2}) x + \frac{5}{6} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Find the lowest common denominator:

$(\frac{7}{8} + \frac{(1 \cdot 4)}{(2 \cdot 4)}) x + \frac{5}{6} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Multiply the denominators:

$(\frac{7}{8} + \frac{(1 \cdot 4)}{8}) x + \frac{5}{6} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Multiply the numerators:

$(\frac{7}{8} + \frac{4}{8}) x + \frac{5}{6} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Combine the fractions:

$\frac{(7 + 4)}{8} \cdot x + \frac{5}{6} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Combine the numerators:

$\frac{11}{8} \cdot x + \frac{5}{6} = (\frac{- 1}{2} \cdot x - 5) + \frac{1}{2} x$

Group like terms:

$\frac{11}{8} \cdot x + \frac{5}{6} = (\frac{- 1}{2} \cdot x + \frac{1}{2} x) - 5$

Combine the fractions:

$\frac{11}{8} \cdot x + \frac{5}{6} = \frac{(- 1 + 1)}{2} x - 5$

Combine the numerators:

$\frac{11}{8} \cdot x + \frac{5}{6} = \frac{0}{2} x - 5$

Reduce the zero numerator:

$\frac{11}{8} x + \frac{5}{6} = 0 x - 5$

Simplify the arithmetic:

$\frac{11}{8} x + \frac{5}{6} = - 5$

Subtract from both sides:

$(\frac{11}{8} x + \frac{5}{6}) - \frac{5}{6} = - 5 - \frac{5}{6}$

Combine the fractions:

$\frac{11}{8} x + \frac{(5 - 5)}{6} = - 5 - \frac{5}{6}$

Combine the numerators:

$\frac{11}{8} x + \frac{0}{6} = - 5 - \frac{5}{6}$

Reduce the zero numerator:

$\frac{11}{8} x + 0 = - 5 - \frac{5}{6}$

Simplify the arithmetic:

$\frac{11}{8} x = - 5 - \frac{5}{6}$

Convert the integer into a fraction:

$\frac{11}{8} x = \frac{- 30}{6} + \frac{- 5}{6}$

Combine the fractions:

$\frac{11}{8} x = \frac{(- 30 - 5)}{6}$

Combine the numerators:

$\frac{11}{8} x = \frac{- 35}{6}$

Multiply both sides by inverse fraction :

$(\frac{11}{8} x) \cdot \frac{8}{11} = (\frac{- 35}{6}) \cdot \frac{8}{11}$

Group like terms:

$(\frac{11}{8} \cdot \frac{8}{11}) x = (\frac{- 35}{6}) \cdot \frac{8}{11}$

Multiply the coefficients:

$\frac{(11 \cdot 8)}{(8 \cdot 11)} x = (\frac{- 35}{6}) \cdot \frac{8}{11}$

Simplify the fraction:

$x = (\frac{- 35}{6}) \cdot \frac{8}{11}$

Multiply the fraction(s):

$x = \frac{(- 35 \cdot 8)}{(6 \cdot 11)}$

Simplify the arithmetic:

$x = \frac{- 140}{(3 \cdot 11)}$

$x = \frac{- 140}{33}$

3. List the solutions

$x = \frac{100}{9}, - \frac{140}{33}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{7}{8} x + \frac{5}{6} |$
$y = | \frac{1}{2} x + 5 |$
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 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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