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Solution - Absolute value equations

Exact form:

w = - 6, \frac{63}{4}

w=-6 , \frac{63}{4}

Mixed number form:

w = - 6, 15 \frac{3}{4}

w=-6 , 15\frac{3}{4}

Decimal form:

w = - 6, 15.75

w=-6 , 15.75

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
$| \frac{1}{9} w - 9 | = | \frac{7}{9} w - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| \frac{1}{9} w - 9 \| = \| \frac{7}{9} w - 5 \|$
$x = + y$	$(\frac{1}{9} w - 9) = (\frac{7}{9} w - 5)$
$x = - y$	$(\frac{1}{9} w - 9) = - (\frac{7}{9} w - 5)$
$+ x = y$	$(\frac{1}{9} w - 9) = (\frac{7}{9} w - 5)$
$- x = y$	$- (\frac{1}{9} w - 9) = (\frac{7}{9} w - 5)$

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:

$\| x \| = \| y \|$	$\| \frac{1}{9} w - 9 \| = \| \frac{7}{9} w - 5 \|$
$x = + y$ , $+ x = y$	$(\frac{1}{9} w - 9) = (\frac{7}{9} w - 5)$
$x = - y$ , $- x = y$	$(\frac{1}{9} w - 9) = - (\frac{7}{9} w - 5)$

2. Solve the two equations for w

22 additional steps

$(\frac{1}{9} \cdot w - 9) = (\frac{7}{9} w - 5)$

Subtract from both sides:

$(\frac{1}{9} w - 9) - \frac{7}{9} \cdot w = (\frac{7}{9} w - 5) - \frac{7}{9} w$

Group like terms:

$(\frac{1}{9} \cdot w + \frac{- 7}{9} \cdot w) - 9 = (\frac{7}{9} \cdot w - 5) - \frac{7}{9} w$

Combine the fractions:

$\frac{(1 - 7)}{9} \cdot w - 9 = (\frac{7}{9} \cdot w - 5) - \frac{7}{9} w$

Combine the numerators:

$\frac{- 6}{9} \cdot w - 9 = (\frac{7}{9} \cdot w - 5) - \frac{7}{9} w$

Find the greatest common factor of the numerator and denominator:

$\frac{(- 2 \cdot 3)}{(3 \cdot 3)} \cdot w - 9 = (\frac{7}{9} \cdot w - 5) - \frac{7}{9} w$

Factor out and cancel the greatest common factor:

$\frac{- 2}{3} \cdot w - 9 = (\frac{7}{9} \cdot w - 5) - \frac{7}{9} w$

Group like terms:

$\frac{- 2}{3} \cdot w - 9 = (\frac{7}{9} \cdot w + \frac{- 7}{9} w) - 5$

Combine the fractions:

$\frac{- 2}{3} \cdot w - 9 = \frac{(7 - 7)}{9} w - 5$

Combine the numerators:

$\frac{- 2}{3} \cdot w - 9 = \frac{0}{9} w - 5$

Reduce the zero numerator:

$\frac{- 2}{3} w - 9 = 0 w - 5$

Simplify the arithmetic:

$\frac{- 2}{3} w - 9 = - 5$

Add to both sides:

$(\frac{- 2}{3} w - 9) + 9 = - 5 + 9$

Simplify the arithmetic:

$\frac{- 2}{3} w = - 5 + 9$

Simplify the arithmetic:

$\frac{- 2}{3} w = 4$

Multiply both sides by inverse fraction :

$(\frac{- 2}{3} w) \cdot \frac{3}{- 2} = 4 \cdot \frac{3}{- 2}$

Move the negative sign from the denominator to the numerator:

$\frac{- 2}{3} w \cdot \frac{- 3}{2} = 4 \cdot \frac{3}{- 2}$

Group like terms:

$(\frac{- 2}{3} \cdot \frac{- 3}{2}) w = 4 \cdot \frac{3}{- 2}$

Multiply the coefficients:

$\frac{(- 2 \cdot - 3)}{(3 \cdot 2)} w = 4 \cdot \frac{3}{- 2}$

Simplify the arithmetic:

$1 w = 4 \cdot \frac{3}{- 2}$

$w = 4 \cdot \frac{3}{- 2}$

Move the negative sign from the denominator to the numerator:

$w = 4 \cdot \frac{- 3}{2}$

Multiply the fraction(s):

$w = \frac{(4 \cdot - 3)}{2}$

Simplify the arithmetic:

$w = - 6$

18 additional steps

$(\frac{1}{9} w - 9) = - (\frac{7}{9} w - 5)$

Expand the parentheses:

$(\frac{1}{9} \cdot w - 9) = \frac{- 7}{9} w + 5$

Add to both sides:

$(\frac{1}{9} w - 9) + \frac{7}{9} \cdot w = (\frac{- 7}{9} w + 5) + \frac{7}{9} w$

Group like terms:

$(\frac{1}{9} \cdot w + \frac{7}{9} \cdot w) - 9 = (\frac{- 7}{9} \cdot w + 5) + \frac{7}{9} w$

Combine the fractions:

$\frac{(1 + 7)}{9} \cdot w - 9 = (\frac{- 7}{9} \cdot w + 5) + \frac{7}{9} w$

Combine the numerators:

$\frac{8}{9} \cdot w - 9 = (\frac{- 7}{9} \cdot w + 5) + \frac{7}{9} w$

Group like terms:

$\frac{8}{9} \cdot w - 9 = (\frac{- 7}{9} \cdot w + \frac{7}{9} w) + 5$

Combine the fractions:

$\frac{8}{9} \cdot w - 9 = \frac{(- 7 + 7)}{9} w + 5$

Combine the numerators:

$\frac{8}{9} \cdot w - 9 = \frac{0}{9} w + 5$

Reduce the zero numerator:

$\frac{8}{9} w - 9 = 0 w + 5$

Simplify the arithmetic:

$\frac{8}{9} w - 9 = 5$

Add to both sides:

$(\frac{8}{9} w - 9) + 9 = 5 + 9$

Simplify the arithmetic:

$\frac{8}{9} w = 5 + 9$

Simplify the arithmetic:

$\frac{8}{9} w = 14$

Multiply both sides by inverse fraction :

$(\frac{8}{9} w) \cdot \frac{9}{8} = 14 \cdot \frac{9}{8}$

Group like terms:

$(\frac{8}{9} \cdot \frac{9}{8}) w = 14 \cdot \frac{9}{8}$

Multiply the coefficients:

$\frac{(8 \cdot 9)}{(9 \cdot 8)} w = 14 \cdot \frac{9}{8}$

Simplify the fraction:

$w = 14 \cdot \frac{9}{8}$

Multiply the fraction(s):

$w = \frac{(14 \cdot 9)}{8}$

Simplify the arithmetic:

$w = \frac{63}{4}$

3. List the solutions

$w = - 6, \frac{63}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{1}{9} w - 9 |$
$y = | \frac{7}{9} w - 5 |$
The equation is true where the two lines cross.

How did we do?

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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 w

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 w

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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