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Solution - Absolute value equations

Exact form:

x = - 5, \frac{1}{5}

x=-5 , \frac{1}{5}

Decimal form:

x = - 5, 0.2

x=-5 , 0.2

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
$| \frac{1}{9} x - \frac{1}{6} | = | \frac{1}{6} x + \frac{1}{9} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| \frac{1}{9} x - \frac{1}{6} \| = \| \frac{1}{6} x + \frac{1}{9} \|$
$x = + y$	$(\frac{1}{9} x - \frac{1}{6}) = (\frac{1}{6} x + \frac{1}{9})$
$x = - y$	$(\frac{1}{9} x - \frac{1}{6}) = - (\frac{1}{6} x + \frac{1}{9})$
$+ x = y$	$(\frac{1}{9} x - \frac{1}{6}) = (\frac{1}{6} x + \frac{1}{9})$
$- x = y$	$- (\frac{1}{9} x - \frac{1}{6}) = (\frac{1}{6} x + \frac{1}{9})$

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:

$\| x \| = \| y \|$	$\| \frac{1}{9} x - \frac{1}{6} \| = \| \frac{1}{6} x + \frac{1}{9} \|$
$x = + y$ , $+ x = y$	$(\frac{1}{9} x - \frac{1}{6}) = (\frac{1}{6} x + \frac{1}{9})$
$x = - y$ , $- x = y$	$(\frac{1}{9} x - \frac{1}{6}) = - (\frac{1}{6} x + \frac{1}{9})$

2. Solve the two equations for x

29 additional steps

$(\frac{1}{9} \cdot x + \frac{- 1}{6}) = (\frac{1}{6} x + \frac{1}{9})$

Subtract from both sides:

$(\frac{1}{9} x + \frac{- 1}{6}) - \frac{1}{6} \cdot x = (\frac{1}{6} x + \frac{1}{9}) - \frac{1}{6} x$

Group like terms:

$(\frac{1}{9} \cdot x + \frac{- 1}{6} \cdot x) + \frac{- 1}{6} = (\frac{1}{6} \cdot x + \frac{1}{9}) - \frac{1}{6} x$

Group the coefficients:

$(\frac{1}{9} + \frac{- 1}{6}) x + \frac{- 1}{6} = (\frac{1}{6} \cdot x + \frac{1}{9}) - \frac{1}{6} x$

Find the lowest common denominator:

$(\frac{(1 \cdot 2)}{(9 \cdot 2)} + \frac{(- 1 \cdot 3)}{(6 \cdot 3)}) x + \frac{- 1}{6} = (\frac{1}{6} \cdot x + \frac{1}{9}) - \frac{1}{6} x$

Multiply the denominators:

$(\frac{(1 \cdot 2)}{18} + \frac{(- 1 \cdot 3)}{18}) x + \frac{- 1}{6} = (\frac{1}{6} \cdot x + \frac{1}{9}) - \frac{1}{6} x$

Multiply the numerators:

$(\frac{2}{18} + \frac{- 3}{18}) x + \frac{- 1}{6} = (\frac{1}{6} \cdot x + \frac{1}{9}) - \frac{1}{6} x$

Combine the fractions:

$\frac{(2 - 3)}{18} \cdot x + \frac{- 1}{6} = (\frac{1}{6} \cdot x + \frac{1}{9}) - \frac{1}{6} x$

Combine the numerators:

$\frac{- 1}{18} \cdot x + \frac{- 1}{6} = (\frac{1}{6} \cdot x + \frac{1}{9}) - \frac{1}{6} x$

Group like terms:

$\frac{- 1}{18} \cdot x + \frac{- 1}{6} = (\frac{1}{6} \cdot x + \frac{- 1}{6} x) + \frac{1}{9}$

Combine the fractions:

$\frac{- 1}{18} \cdot x + \frac{- 1}{6} = \frac{(1 - 1)}{6} x + \frac{1}{9}$

Combine the numerators:

$\frac{- 1}{18} \cdot x + \frac{- 1}{6} = \frac{0}{6} x + \frac{1}{9}$

Reduce the zero numerator:

$\frac{- 1}{18} x + \frac{- 1}{6} = 0 x + \frac{1}{9}$

Simplify the arithmetic:

$\frac{- 1}{18} x + \frac{- 1}{6} = \frac{1}{9}$

Add to both sides:

$(\frac{- 1}{18} x + \frac{- 1}{6}) + \frac{1}{6} = (\frac{1}{9}) + \frac{1}{6}$

Combine the fractions:

$\frac{- 1}{18} x + \frac{(- 1 + 1)}{6} = (\frac{1}{9}) + \frac{1}{6}$

Combine the numerators:

$\frac{- 1}{18} x + \frac{0}{6} = (\frac{1}{9}) + \frac{1}{6}$

Reduce the zero numerator:

$\frac{- 1}{18} x + 0 = (\frac{1}{9}) + \frac{1}{6}$

Simplify the arithmetic:

$\frac{- 1}{18} x = (\frac{1}{9}) + \frac{1}{6}$

Find the lowest common denominator:

$\frac{- 1}{18} x = \frac{(1 \cdot 2)}{(9 \cdot 2)} + \frac{(1 \cdot 3)}{(6 \cdot 3)}$

Multiply the denominators:

$\frac{- 1}{18} x = \frac{(1 \cdot 2)}{18} + \frac{(1 \cdot 3)}{18}$

Multiply the numerators:

$\frac{- 1}{18} x = \frac{2}{18} + \frac{3}{18}$

Combine the fractions:

$\frac{- 1}{18} x = \frac{(2 + 3)}{18}$

Combine the numerators:

$\frac{- 1}{18} x = \frac{5}{18}$

Multiply both sides by inverse fraction :

$(\frac{- 1}{18} x) \cdot \frac{18}{- 1} = (\frac{5}{18}) \cdot \frac{18}{- 1}$

Group like terms:

$(\frac{- 1}{18} \cdot - 18) x = (\frac{5}{18}) \cdot \frac{18}{- 1}$

Multiply the coefficients:

$\frac{(- 1 \cdot - 18)}{18} x = (\frac{5}{18}) \cdot \frac{18}{- 1}$

Simplify the arithmetic:

$1 x = (\frac{5}{18}) \cdot \frac{18}{- 1}$

$x = (\frac{5}{18}) \cdot \frac{18}{- 1}$

Multiply the fraction(s):

$x = \frac{(5 \cdot - 18)}{18}$

Simplify the arithmetic:

$x = - 5$

29 additional steps

$(\frac{1}{9} x + \frac{- 1}{6}) = - (\frac{1}{6} x + \frac{1}{9})$

Expand the parentheses:

$(\frac{1}{9} \cdot x + \frac{- 1}{6}) = \frac{- 1}{6} x + \frac{- 1}{9}$

Add to both sides:

$(\frac{1}{9} x + \frac{- 1}{6}) + \frac{1}{6} \cdot x = (\frac{- 1}{6} x + \frac{- 1}{9}) + \frac{1}{6} x$

Group like terms:

$(\frac{1}{9} \cdot x + \frac{1}{6} \cdot x) + \frac{- 1}{6} = (\frac{- 1}{6} \cdot x + \frac{- 1}{9}) + \frac{1}{6} x$

Group the coefficients:

$(\frac{1}{9} + \frac{1}{6}) x + \frac{- 1}{6} = (\frac{- 1}{6} \cdot x + \frac{- 1}{9}) + \frac{1}{6} x$

Find the lowest common denominator:

$(\frac{(1 \cdot 2)}{(9 \cdot 2)} + \frac{(1 \cdot 3)}{(6 \cdot 3)}) x + \frac{- 1}{6} = (\frac{- 1}{6} \cdot x + \frac{- 1}{9}) + \frac{1}{6} x$

Multiply the denominators:

$(\frac{(1 \cdot 2)}{18} + \frac{(1 \cdot 3)}{18}) x + \frac{- 1}{6} = (\frac{- 1}{6} \cdot x + \frac{- 1}{9}) + \frac{1}{6} x$

Multiply the numerators:

$(\frac{2}{18} + \frac{3}{18}) x + \frac{- 1}{6} = (\frac{- 1}{6} \cdot x + \frac{- 1}{9}) + \frac{1}{6} x$

Combine the fractions:

$\frac{(2 + 3)}{18} \cdot x + \frac{- 1}{6} = (\frac{- 1}{6} \cdot x + \frac{- 1}{9}) + \frac{1}{6} x$

Combine the numerators:

$\frac{5}{18} \cdot x + \frac{- 1}{6} = (\frac{- 1}{6} \cdot x + \frac{- 1}{9}) + \frac{1}{6} x$

Group like terms:

$\frac{5}{18} \cdot x + \frac{- 1}{6} = (\frac{- 1}{6} \cdot x + \frac{1}{6} x) + \frac{- 1}{9}$

Combine the fractions:

$\frac{5}{18} \cdot x + \frac{- 1}{6} = \frac{(- 1 + 1)}{6} x + \frac{- 1}{9}$

Combine the numerators:

$\frac{5}{18} \cdot x + \frac{- 1}{6} = \frac{0}{6} x + \frac{- 1}{9}$

Reduce the zero numerator:

$\frac{5}{18} x + \frac{- 1}{6} = 0 x + \frac{- 1}{9}$

Simplify the arithmetic:

$\frac{5}{18} x + \frac{- 1}{6} = \frac{- 1}{9}$

Add to both sides:

$(\frac{5}{18} x + \frac{- 1}{6}) + \frac{1}{6} = (\frac{- 1}{9}) + \frac{1}{6}$

Combine the fractions:

$\frac{5}{18} x + \frac{(- 1 + 1)}{6} = (\frac{- 1}{9}) + \frac{1}{6}$

Combine the numerators:

$\frac{5}{18} x + \frac{0}{6} = (\frac{- 1}{9}) + \frac{1}{6}$

Reduce the zero numerator:

$\frac{5}{18} x + 0 = (\frac{- 1}{9}) + \frac{1}{6}$

Simplify the arithmetic:

$\frac{5}{18} x = (\frac{- 1}{9}) + \frac{1}{6}$

Find the lowest common denominator:

$\frac{5}{18} x = \frac{(- 1 \cdot 2)}{(9 \cdot 2)} + \frac{(1 \cdot 3)}{(6 \cdot 3)}$

Multiply the denominators:

$\frac{5}{18} x = \frac{(- 1 \cdot 2)}{18} + \frac{(1 \cdot 3)}{18}$

Multiply the numerators:

$\frac{5}{18} x = \frac{- 2}{18} + \frac{3}{18}$

Combine the fractions:

$\frac{5}{18} x = \frac{(- 2 + 3)}{18}$

Combine the numerators:

$\frac{5}{18} x = \frac{1}{18}$

Multiply both sides by inverse fraction :

$(\frac{5}{18} x) \cdot \frac{18}{5} = (\frac{1}{18}) \cdot \frac{18}{5}$

Group like terms:

$(\frac{5}{18} \cdot \frac{18}{5}) x = (\frac{1}{18}) \cdot \frac{18}{5}$

Multiply the coefficients:

$\frac{(5 \cdot 18)}{(18 \cdot 5)} x = (\frac{1}{18}) \cdot \frac{18}{5}$

Simplify the fraction:

$x = (\frac{1}{18}) \cdot \frac{18}{5}$

Multiply the fraction(s):

$x = \frac{(1 \cdot 18)}{(18 \cdot 5)}$

Simplify the arithmetic:

$x = \frac{1}{5}$

3. List the solutions

$x = - 5, \frac{1}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{1}{9} x - \frac{1}{6} |$
$y = | \frac{1}{6} x + \frac{1}{9} |$
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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