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Solution - Absolute value equations

Exact form:

w = - 12, 4

w=-12 , 4

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
$| \frac{1}{2} w - 6 | = | w |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| \frac{1}{2} w - 6 \| = \| w \|$
$x = + y$	$(\frac{1}{2} w - 6) = (w)$
$x = - y$	$(\frac{1}{2} w - 6) = - (w)$
$+ x = y$	$(\frac{1}{2} w - 6) = (w)$
$- x = y$	$- (\frac{1}{2} w - 6) = (w)$

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:

$\| x \| = \| y \|$	$\| \frac{1}{2} w - 6 \| = \| w \|$
$x = + y$ , $+ x = y$	$(\frac{1}{2} w - 6) = (w)$
$x = - y$ , $- x = y$	$(\frac{1}{2} w - 6) = - (w)$

2. Solve the two equations for w

15 additional steps

$(\frac{1}{2} w - 6) = w$

Subtract from both sides:

$(\frac{1}{2} w - 6) - w = w - w$

Group like terms:

$(\frac{1}{2} w - w) - 6 = w - w$

Group the coefficients:

$(\frac{1}{2} - 1) w - 6 = w - w$

Convert the integer into a fraction:

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

Combine the fractions:

$\frac{(1 - 2)}{2} w - 6 = w - w$

Combine the numerators:

$\frac{- 1}{2} w - 6 = w - w$

Simplify the arithmetic:

$\frac{- 1}{2} w - 6 = 0$

Add to both sides:

$(\frac{- 1}{2} w - 6) + 6 = 0 + 6$

Simplify the arithmetic:

$\frac{- 1}{2} w = 0 + 6$

Simplify the arithmetic:

$\frac{- 1}{2} w = 6$

Multiply both sides by inverse fraction :

$(\frac{- 1}{2} w) \cdot \frac{2}{- 1} = 6 \cdot \frac{2}{- 1}$

Group like terms:

$(\frac{- 1}{2} \cdot - 2) w = 6 \cdot \frac{2}{- 1}$

Multiply the coefficients:

$\frac{(- 1 \cdot - 2)}{2} w = 6 \cdot \frac{2}{- 1}$

Simplify the arithmetic:

$1 w = 6 \cdot \frac{2}{- 1}$

$w = 6 \cdot \frac{2}{- 1}$

Simplify the arithmetic:

$w = - 12$

15 additional steps

$(\frac{1}{2} w - 6) = - w$

Add to both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

$\frac{3}{2} w - 6 = - w + w$

Simplify the arithmetic:

$\frac{3}{2} w - 6 = 0$

Add to both sides:

$(\frac{3}{2} w - 6) + 6 = 0 + 6$

Simplify the arithmetic:

$\frac{3}{2} w = 0 + 6$

Simplify the arithmetic:

$\frac{3}{2} w = 6$

Multiply both sides by inverse fraction :

$(\frac{3}{2} w) \cdot \frac{2}{3} = 6 \cdot \frac{2}{3}$

Group like terms:

$(\frac{3}{2} \cdot \frac{2}{3}) w = 6 \cdot \frac{2}{3}$

Multiply the coefficients:

$\frac{(3 \cdot 2)}{(2 \cdot 3)} w = 6 \cdot \frac{2}{3}$

Simplify the fraction:

$w = 6 \cdot \frac{2}{3}$

Multiply the fraction(s):

$w = \frac{(6 \cdot 2)}{3}$

Simplify the arithmetic:

$w = 4$

3. List the solutions

$w = - 12, 4$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{1}{2} w - 6 |$
$y = | w |$
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 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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