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Solution - Absolute value equations

Exact form:

w = - \frac{4}{3}, \frac{4}{21}

w=-\frac{4}{3} , \frac{4}{21}

Mixed number form:

w = - 1 \frac{1}{3}, \frac{4}{21}

w=-1\frac{1}{3} , \frac{4}{21}

Decimal form:

w = - 1.333, 0.190

w=-1.333 , 0.190

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
$| 9 w - 4 | = | 12 w |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 w - 4 \| = \| 12 w \|$
$x = + y$	$(9 w - 4) = (12 w)$
$x = - y$	$(9 w - 4) = - (12 w)$
$+ x = y$	$(9 w - 4) = (12 w)$
$- x = y$	$- (9 w - 4) = (12 w)$

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:

$\| x \| = \| y \|$	$\| 9 w - 4 \| = \| 12 w \|$
$x = + y$ , $+ x = y$	$(9 w - 4) = (12 w)$
$x = - y$ , $- x = y$	$(9 w - 4) = - (12 w)$

2. Solve the two equations for w

10 additional steps

$(9 w - 4) = 12 w$

Subtract from both sides:

$(9 w - 4) - 12 w = (12 w) - 12 w$

Group like terms:

$(9 w - 12 w) - 4 = (12 w) - 12 w$

Simplify the arithmetic:

$- 3 w - 4 = (12 w) - 12 w$

Simplify the arithmetic:

$- 3 w - 4 = 0$

Add to both sides:

$(- 3 w - 4) + 4 = 0 + 4$

Simplify the arithmetic:

$- 3 w = 0 + 4$

Simplify the arithmetic:

$- 3 w = 4$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

7 additional steps

$(9 w - 4) = - 12 w$

Add to both sides:

$(9 w - 4) + 4 = (- 12 w) + 4$

Simplify the arithmetic:

$9 w = (- 12 w) + 4$

Add to both sides:

$(9 w) + 12 w = ((- 12 w) + 4) + 12 w$

Simplify the arithmetic:

$21 w = ((- 12 w) + 4) + 12 w$

Group like terms:

$21 w = (- 12 w + 12 w) + 4$

Simplify the arithmetic:

$21 w = 4$

Divide both sides by :

$\frac{(21 w)}{21} = \frac{4}{21}$

Simplify the fraction:

$w = \frac{4}{21}$

3. List the solutions

$w = - \frac{4}{3}, \frac{4}{21}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 w - 4 |$
$y = | 12 w |$
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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