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Solution - Absolute value equations

Exact form:

w = \frac{6}{11}, \frac{6}{13}

w=\frac{6}{11} , \frac{6}{13}

Decimal form:

w = 0.545, 0.462

w=0.545 , 0.462

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
$| 12 w - 6 | = | w |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 12 w - 6 \| = \| w \|$
$x = + y$	$(12 w - 6) = (w)$
$x = - y$	$(12 w - 6) = - (w)$
$+ x = y$	$(12 w - 6) = (w)$
$- x = y$	$- (12 w - 6) = (w)$

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:

$\| x \| = \| y \|$	$\| 12 w - 6 \| = \| w \|$
$x = + y$ , $+ x = y$	$(12 w - 6) = (w)$
$x = - y$ , $- x = y$	$(12 w - 6) = - (w)$

2. Solve the two equations for w

8 additional steps

$(12 w - 6) = w$

Subtract from both sides:

$(12 w - 6) - w = w - w$

Group like terms:

$(12 w - w) - 6 = w - w$

Simplify the arithmetic:

$11 w - 6 = w - w$

Simplify the arithmetic:

$11 w - 6 = 0$

Add to both sides:

$(11 w - 6) + 6 = 0 + 6$

Simplify the arithmetic:

$11 w = 0 + 6$

Simplify the arithmetic:

$11 w = 6$

Divide both sides by :

$\frac{(11 w)}{11} = \frac{6}{11}$

Simplify the fraction:

$w = \frac{6}{11}$

8 additional steps

$(12 w - 6) = - w$

Add to both sides:

$(12 w - 6) + w = - w + w$

Group like terms:

$(12 w + w) - 6 = - w + w$

Simplify the arithmetic:

$13 w - 6 = - w + w$

Simplify the arithmetic:

$13 w - 6 = 0$

Add to both sides:

$(13 w - 6) + 6 = 0 + 6$

Simplify the arithmetic:

$13 w = 0 + 6$

Simplify the arithmetic:

$13 w = 6$

Divide both sides by :

$\frac{(13 w)}{13} = \frac{6}{13}$

Simplify the fraction:

$w = \frac{6}{13}$

3. List the solutions

$w = \frac{6}{11}, \frac{6}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 12 w - 6 |$
$y = | w |$
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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