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Solution - Absolute value equations

Exact form:

n = \frac{12}{13}, - \frac{72}{7}

n=\frac{12}{13} , -\frac{72}{7}

Mixed number form:

n = \frac{12}{13}, - 10 \frac{2}{7}

n=\frac{12}{13} , -10\frac{2}{7}

Decimal form:

n = 0.923, - 10.286

n=0.923 , -10.286

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
$| - \frac{1}{2} n + 7 | = | \frac{5}{3} n + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - \frac{1}{2} n + 7 \| = \| \frac{5}{3} n + 5 \|$
$x = + y$	$(- \frac{1}{2} n + 7) = (\frac{5}{3} n + 5)$
$x = - y$	$(- \frac{1}{2} n + 7) = - (\frac{5}{3} n + 5)$
$+ x = y$	$(- \frac{1}{2} n + 7) = (\frac{5}{3} n + 5)$
$- x = y$	$- (- \frac{1}{2} n + 7) = (\frac{5}{3} n + 5)$

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:

$\| x \| = \| y \|$	$\| - \frac{1}{2} n + 7 \| = \| \frac{5}{3} n + 5 \|$
$x = + y$ , $+ x = y$	$(- \frac{1}{2} n + 7) = (\frac{5}{3} n + 5)$
$x = - y$ , $- x = y$	$(- \frac{1}{2} n + 7) = - (\frac{5}{3} n + 5)$

2. Solve the two equations for n

24 additional steps

$(\frac{- 1}{2} \cdot n + 7) = (\frac{5}{3} n + 5)$

Subtract from both sides:

$(\frac{- 1}{2} n + 7) - \frac{5}{3} \cdot n = (\frac{5}{3} n + 5) - \frac{5}{3} n$

Group like terms:

$(\frac{- 1}{2} \cdot n + \frac{- 5}{3} \cdot n) + 7 = (\frac{5}{3} \cdot n + 5) - \frac{5}{3} n$

Group the coefficients:

$(\frac{- 1}{2} + \frac{- 5}{3}) n + 7 = (\frac{5}{3} \cdot n + 5) - \frac{5}{3} n$

Find the lowest common denominator:

$(\frac{(- 1 \cdot 3)}{(2 \cdot 3)} + \frac{(- 5 \cdot 2)}{(3 \cdot 2)}) n + 7 = (\frac{5}{3} \cdot n + 5) - \frac{5}{3} n$

Multiply the denominators:

$(\frac{(- 1 \cdot 3)}{6} + \frac{(- 5 \cdot 2)}{6}) n + 7 = (\frac{5}{3} \cdot n + 5) - \frac{5}{3} n$

Multiply the numerators:

$(\frac{- 3}{6} + \frac{- 10}{6}) n + 7 = (\frac{5}{3} \cdot n + 5) - \frac{5}{3} n$

Combine the fractions:

$\frac{(- 3 - 10)}{6} \cdot n + 7 = (\frac{5}{3} \cdot n + 5) - \frac{5}{3} n$

Combine the numerators:

$\frac{- 13}{6} \cdot n + 7 = (\frac{5}{3} \cdot n + 5) - \frac{5}{3} n$

Group like terms:

$\frac{- 13}{6} \cdot n + 7 = (\frac{5}{3} \cdot n + \frac{- 5}{3} n) + 5$

Combine the fractions:

$\frac{- 13}{6} \cdot n + 7 = \frac{(5 - 5)}{3} n + 5$

Combine the numerators:

$\frac{- 13}{6} \cdot n + 7 = \frac{0}{3} n + 5$

Reduce the zero numerator:

$\frac{- 13}{6} n + 7 = 0 n + 5$

Simplify the arithmetic:

$\frac{- 13}{6} n + 7 = 5$

Subtract from both sides:

$(\frac{- 13}{6} n + 7) - 7 = 5 - 7$

Simplify the arithmetic:

$\frac{- 13}{6} n = 5 - 7$

Simplify the arithmetic:

$\frac{- 13}{6} n = - 2$

Multiply both sides by inverse fraction :

$(\frac{- 13}{6} n) \cdot \frac{6}{- 13} = - 2 \cdot \frac{6}{- 13}$

Move the negative sign from the denominator to the numerator:

$\frac{- 13}{6} n \cdot \frac{- 6}{13} = - 2 \cdot \frac{6}{- 13}$

Group like terms:

$(\frac{- 13}{6} \cdot \frac{- 6}{13}) n = - 2 \cdot \frac{6}{- 13}$

Multiply the coefficients:

$\frac{(- 13 \cdot - 6)}{(6 \cdot 13)} n = - 2 \cdot \frac{6}{- 13}$

Simplify the arithmetic:

$1 n = - 2 \cdot \frac{6}{- 13}$

$n = - 2 \cdot \frac{6}{- 13}$

Move the negative sign from the denominator to the numerator:

$n = - 2 \cdot \frac{- 6}{13}$

Multiply the fraction(s):

$n = \frac{(- 2 \cdot - 6)}{13}$

Simplify the arithmetic:

$n = \frac{12}{13}$

22 additional steps

$(\frac{- 1}{2} n + 7) = - (\frac{5}{3} n + 5)$

Expand the parentheses:

$(\frac{- 1}{2} \cdot n + 7) = \frac{- 5}{3} n - 5$

Add to both sides:

$(\frac{- 1}{2} n + 7) + \frac{5}{3} \cdot n = (\frac{- 5}{3} n - 5) + \frac{5}{3} n$

Group like terms:

$(\frac{- 1}{2} \cdot n + \frac{5}{3} \cdot n) + 7 = (\frac{- 5}{3} \cdot n - 5) + \frac{5}{3} n$

Group the coefficients:

$(\frac{- 1}{2} + \frac{5}{3}) n + 7 = (\frac{- 5}{3} \cdot n - 5) + \frac{5}{3} n$

Find the lowest common denominator:

$(\frac{(- 1 \cdot 3)}{(2 \cdot 3)} + \frac{(5 \cdot 2)}{(3 \cdot 2)}) n + 7 = (\frac{- 5}{3} \cdot n - 5) + \frac{5}{3} n$

Multiply the denominators:

$(\frac{(- 1 \cdot 3)}{6} + \frac{(5 \cdot 2)}{6}) n + 7 = (\frac{- 5}{3} \cdot n - 5) + \frac{5}{3} n$

Multiply the numerators:

$(\frac{- 3}{6} + \frac{10}{6}) n + 7 = (\frac{- 5}{3} \cdot n - 5) + \frac{5}{3} n$

Combine the fractions:

$\frac{(- 3 + 10)}{6} \cdot n + 7 = (\frac{- 5}{3} \cdot n - 5) + \frac{5}{3} n$

Combine the numerators:

$\frac{7}{6} \cdot n + 7 = (\frac{- 5}{3} \cdot n - 5) + \frac{5}{3} n$

Group like terms:

$\frac{7}{6} \cdot n + 7 = (\frac{- 5}{3} \cdot n + \frac{5}{3} n) - 5$

Combine the fractions:

$\frac{7}{6} \cdot n + 7 = \frac{(- 5 + 5)}{3} n - 5$

Combine the numerators:

$\frac{7}{6} \cdot n + 7 = \frac{0}{3} n - 5$

Reduce the zero numerator:

$\frac{7}{6} n + 7 = 0 n - 5$

Simplify the arithmetic:

$\frac{7}{6} n + 7 = - 5$

Subtract from both sides:

$(\frac{7}{6} n + 7) - 7 = - 5 - 7$

Simplify the arithmetic:

$\frac{7}{6} n = - 5 - 7$

Simplify the arithmetic:

$\frac{7}{6} n = - 12$

Multiply both sides by inverse fraction :

$(\frac{7}{6} n) \cdot \frac{6}{7} = - 12 \cdot \frac{6}{7}$

Group like terms:

$(\frac{7}{6} \cdot \frac{6}{7}) n = - 12 \cdot \frac{6}{7}$

Multiply the coefficients:

$\frac{(7 \cdot 6)}{(6 \cdot 7)} n = - 12 \cdot \frac{6}{7}$

Simplify the fraction:

$n = - 12 \cdot \frac{6}{7}$

Multiply the fraction(s):

$n = \frac{(- 12 \cdot 6)}{7}$

Simplify the arithmetic:

$n = \frac{- 72}{7}$

3. List the solutions

$n = \frac{12}{13}, - \frac{72}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - \frac{1}{2} n + 7 |$
$y = | \frac{5}{3} n + 5 |$
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 n

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 n

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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