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Solution - Absolute value equations

Exact form:

x = - 66, - \frac{54}{5}

x=-66 , -\frac{54}{5}

Mixed number form:

x = - 66, - 10 \frac{4}{5}

x=-66 , -10\frac{4}{5}

Decimal form:

x = - 66, - 10.8

x=-66 , -10.8

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
$\frac{1}{3} | x - 3 | = \frac{1}{2} | x + 20 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\frac{1}{3} \| x - 3 \| = \frac{1}{2} \| x + 20 \|$
$x = + y$	$\frac{1}{3} (x - 3) = \frac{1}{2} (x + 20)$
$x = - y$	$\frac{1}{3} (x - 3) = \frac{1}{2} (- (x + 20))$
$+ x = y$	$\frac{1}{3} (x - 3) = \frac{1}{2} (x + 20)$
$- x = y$	$\frac{1}{3} (- (x - 3)) = \frac{1}{2} (x + 20)$

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:

$\| x \| = \| y \|$	$\frac{1}{3} \| x - 3 \| = \frac{1}{2} \| x + 20 \|$
$x = + y$ , $+ x = y$	$\frac{1}{3} (x - 3) = \frac{1}{2} (x + 20)$
$x = - y$ , $- x = y$	$\frac{1}{3} (x - 3) = \frac{1}{2} (- (x + 20))$

2. Solve the two equations for x

29 additional steps

$\frac{1}{3} \cdot (x - 3) = \frac{1}{2} \cdot (x + 20)$

Multiply the fraction(s):

$\frac{(1 \cdot (x - 3))}{3} = \frac{1}{2} \cdot (x + 20)$

Break up the fraction:

$\frac{x}{3} + \frac{- 3}{3} = \frac{1}{2} \cdot (x + 20)$

Find the greatest common factor of the numerator and denominator:

$\frac{x}{3} + \frac{(- 1 \cdot 3)}{(1 \cdot 3)} = \frac{1}{2} \cdot (x + 20)$

Factor out and cancel the greatest common factor:

$\frac{x}{3} - 1 = \frac{1}{2} \cdot (x + 20)$

Multiply the fraction(s):

$\frac{x}{3} - 1 = \frac{(1 \cdot (x + 20))}{2}$

Break up the fraction:

$\frac{x}{3} - 1 = \frac{x}{2} + \frac{20}{2}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$\frac{x}{3} - 1 = \frac{x}{2} + 10$

Subtract from both sides:

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

Group like terms:

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

Group the coefficients:

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

Find the lowest common denominator:

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

Multiply the denominators:

$(\frac{(1 \cdot 2)}{6} + \frac{(- 1 \cdot 3)}{6}) x - 1 = (\frac{x}{2} + 10) - \frac{x}{2}$

Multiply the numerators:

$(\frac{2}{6} + \frac{- 3}{6}) x - 1 = (\frac{x}{2} + 10) - \frac{x}{2}$

Combine the fractions:

$\frac{(2 - 3)}{6} x - 1 = (\frac{x}{2} + 10) - \frac{x}{2}$

Combine the numerators:

$\frac{- 1}{6} x - 1 = (\frac{x}{2} + 10) - \frac{x}{2}$

Group like terms:

$\frac{- 1}{6} \cdot x - 1 = (\frac{x}{2} + \frac{- 1}{2} x) + 10$

Combine the fractions:

$\frac{- 1}{6} \cdot x - 1 = \frac{(1 - 1)}{2} x + 10$

Combine the numerators:

$\frac{- 1}{6} \cdot x - 1 = \frac{0}{2} x + 10$

Reduce the zero numerator:

$\frac{- 1}{6} x - 1 = 0 x + 10$

Simplify the arithmetic:

$\frac{- 1}{6} x - 1 = 10$

Add to both sides:

$(\frac{- 1}{6} x - 1) + 1 = 10 + 1$

Simplify the arithmetic:

$\frac{- 1}{6} x = 10 + 1$

Simplify the arithmetic:

$\frac{- 1}{6} x = 11$

Multiply both sides by inverse fraction :

$(\frac{- 1}{6} x) \cdot \frac{6}{- 1} = 11 \cdot \frac{6}{- 1}$

Group like terms:

$(\frac{- 1}{6} \cdot - 6) x = 11 \cdot \frac{6}{- 1}$

Multiply the coefficients:

$\frac{(- 1 \cdot - 6)}{6} x = 11 \cdot \frac{6}{- 1}$

Simplify the arithmetic:

$1 x = 11 \cdot \frac{6}{- 1}$

$x = 11 \cdot \frac{6}{- 1}$

Simplify the arithmetic:

$x = - 66$

30 additional steps

$\frac{1}{3} \cdot (x - 3) = \frac{1}{2} \cdot (- (x + 20))$

Multiply the fraction(s):

$\frac{(1 \cdot (x - 3))}{3} = \frac{1}{2} \cdot (- (x + 20))$

Break up the fraction:

$\frac{x}{3} + \frac{- 3}{3} = \frac{1}{2} \cdot (- (x + 20))$

Find the greatest common factor of the numerator and denominator:

$\frac{x}{3} + \frac{(- 1 \cdot 3)}{(1 \cdot 3)} = \frac{1}{2} \cdot (- (x + 20))$

Factor out and cancel the greatest common factor:

$\frac{x}{3} - 1 = \frac{1}{2} \cdot (- (x + 20))$

Multiply the fraction(s):

$\frac{x}{3} - 1 = \frac{(1 \cdot (- (x + 20)))}{2}$

Expand the parentheses:

$\frac{x}{3} - 1 = \frac{(- x - 20)}{2}$

Break up the fraction:

$\frac{x}{3} - 1 = \frac{- x}{2} + \frac{- 20}{2}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$\frac{x}{3} - 1 = \frac{- x}{2} - 10$

Add to both sides:

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

Group like terms:

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

Group the coefficients:

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

Find the lowest common denominator:

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

Multiply the denominators:

$(\frac{(1 \cdot 2)}{6} + \frac{(1 \cdot 3)}{6}) x - 1 = (\frac{- x}{2} - 10) + \frac{1}{2} x$

Multiply the numerators:

$(\frac{2}{6} + \frac{3}{6}) x - 1 = (\frac{- x}{2} - 10) + \frac{1}{2} x$

Combine the fractions:

$\frac{(2 + 3)}{6} \cdot x - 1 = (\frac{- x}{2} - 10) + \frac{1}{2} x$

Combine the numerators:

$\frac{5}{6} \cdot x - 1 = (\frac{- x}{2} - 10) + \frac{1}{2} x$

Group like terms:

$\frac{5}{6} \cdot x - 1 = (\frac{- x}{2} + \frac{1}{2} x) - 10$

Combine the fractions:

$\frac{5}{6} \cdot x - 1 = \frac{(- 1 + 1)}{2} x - 10$

Combine the numerators:

$\frac{5}{6} \cdot x - 1 = \frac{0}{2} x - 10$

Reduce the zero numerator:

$\frac{5}{6} x - 1 = 0 x - 10$

Simplify the arithmetic:

$\frac{5}{6} x - 1 = - 10$

Add to both sides:

$(\frac{5}{6} x - 1) + 1 = - 10 + 1$

Simplify the arithmetic:

$\frac{5}{6} x = - 10 + 1$

Simplify the arithmetic:

$\frac{5}{6} x = - 9$

Multiply both sides by inverse fraction :

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the fraction:

$x = - 9 \cdot \frac{6}{5}$

Multiply the fraction(s):

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

Simplify the arithmetic:

$x = \frac{- 54}{5}$

3. List the solutions

$x = - 66, - \frac{54}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = \frac{1}{3} | x - 3 |$
$y = \frac{1}{2} | x + 20 |$
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

Latest Related Drills Solved