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Solution - Absolute value equations

Exact form:

x = 45, - \frac{105}{11}

x=45 , -\frac{105}{11}

Mixed number form:

x = 45, - 9 \frac{6}{11}

x=45 , -9\frac{6}{11}

Decimal form:

x = 45, - 9.545

x=45 , -9.545

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

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

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:

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

2. Solve the two equations for x

21 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Group the coefficients:

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

Find the lowest common denominator:

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

Multiply the denominators:

$(\frac{(1 \cdot 5)}{15} + \frac{(- 2 \cdot 3)}{15}) x + 5 = (\frac{2}{5} \cdot x + 2) - \frac{2}{5} x$

Multiply the numerators:

$(\frac{5}{15} + \frac{- 6}{15}) x + 5 = (\frac{2}{5} \cdot x + 2) - \frac{2}{5} x$

Combine the fractions:

$\frac{(5 - 6)}{15} \cdot x + 5 = (\frac{2}{5} \cdot x + 2) - \frac{2}{5} x$

Combine the numerators:

$\frac{- 1}{15} \cdot x + 5 = (\frac{2}{5} \cdot x + 2) - \frac{2}{5} x$

Group like terms:

$\frac{- 1}{15} \cdot x + 5 = (\frac{2}{5} \cdot x + \frac{- 2}{5} x) + 2$

Combine the fractions:

$\frac{- 1}{15} \cdot x + 5 = \frac{(2 - 2)}{5} x + 2$

Combine the numerators:

$\frac{- 1}{15} \cdot x + 5 = \frac{0}{5} x + 2$

Reduce the zero numerator:

$\frac{- 1}{15} x + 5 = 0 x + 2$

Simplify the arithmetic:

$\frac{- 1}{15} x + 5 = 2$

Subtract from both sides:

$(\frac{- 1}{15} x + 5) - 5 = 2 - 5$

Simplify the arithmetic:

$\frac{- 1}{15} x = 2 - 5$

Simplify the arithmetic:

$\frac{- 1}{15} x = - 3$

Multiply both sides by inverse fraction :

$(\frac{- 1}{15} x) \cdot \frac{15}{- 1} = - 3 \cdot \frac{15}{- 1}$

Group like terms:

$(\frac{- 1}{15} \cdot - 15) x = - 3 \cdot \frac{15}{- 1}$

Multiply the coefficients:

$\frac{(- 1 \cdot - 15)}{15} x = - 3 \cdot \frac{15}{- 1}$

Simplify the arithmetic:

$1 x = - 3 \cdot \frac{15}{- 1}$

$x = - 3 \cdot \frac{15}{- 1}$

Simplify the arithmetic:

$x = 45$

22 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Group the coefficients:

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

Find the lowest common denominator:

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

Multiply the denominators:

$(\frac{(1 \cdot 5)}{15} + \frac{(2 \cdot 3)}{15}) x + 5 = (\frac{- 2}{5} \cdot x - 2) + \frac{2}{5} x$

Multiply the numerators:

$(\frac{5}{15} + \frac{6}{15}) x + 5 = (\frac{- 2}{5} \cdot x - 2) + \frac{2}{5} x$

Combine the fractions:

$\frac{(5 + 6)}{15} \cdot x + 5 = (\frac{- 2}{5} \cdot x - 2) + \frac{2}{5} x$

Combine the numerators:

$\frac{11}{15} \cdot x + 5 = (\frac{- 2}{5} \cdot x - 2) + \frac{2}{5} x$

Group like terms:

$\frac{11}{15} \cdot x + 5 = (\frac{- 2}{5} \cdot x + \frac{2}{5} x) - 2$

Combine the fractions:

$\frac{11}{15} \cdot x + 5 = \frac{(- 2 + 2)}{5} x - 2$

Combine the numerators:

$\frac{11}{15} \cdot x + 5 = \frac{0}{5} x - 2$

Reduce the zero numerator:

$\frac{11}{15} x + 5 = 0 x - 2$

Simplify the arithmetic:

$\frac{11}{15} x + 5 = - 2$

Subtract from both sides:

$(\frac{11}{15} x + 5) - 5 = - 2 - 5$

Simplify the arithmetic:

$\frac{11}{15} x = - 2 - 5$

Simplify the arithmetic:

$\frac{11}{15} x = - 7$

Multiply both sides by inverse fraction :

$(\frac{11}{15} x) \cdot \frac{15}{11} = - 7 \cdot \frac{15}{11}$

Group like terms:

$(\frac{11}{15} \cdot \frac{15}{11}) x = - 7 \cdot \frac{15}{11}$

Multiply the coefficients:

$\frac{(11 \cdot 15)}{(15 \cdot 11)} x = - 7 \cdot \frac{15}{11}$

Simplify the fraction:

$x = - 7 \cdot \frac{15}{11}$

Multiply the fraction(s):

$x = \frac{(- 7 \cdot 15)}{11}$

Simplify the arithmetic:

$x = \frac{- 105}{11}$

3. List the solutions

$x = 45, - \frac{105}{11}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{1}{3} x + 5 |$
$y = | \frac{2}{5} x + 2 |$
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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