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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

w = 6, 1 \frac{1}{2}

w=6 , 1\frac{1}{2}

Decimal form:

w = 6, 1.5

w=6 , 1.5

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
$6 | w - 3 | = | 2 w + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$6 \| w - 3 \| = \| 2 w + 6 \|$
$x = + y$	$6 (w - 3) = (2 w + 6)$
$x = - y$	$6 (w - 3) = - (2 w + 6)$
$+ x = y$	$6 (w - 3) = (2 w + 6)$
$- x = y$	$6 (- (w - 3)) = (2 w + 6)$

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:

$\| x \| = \| y \|$	$6 \| w - 3 \| = \| 2 w + 6 \|$
$x = + y$ , $+ x = y$	$6 (w - 3) = (2 w + 6)$
$x = - y$ , $- x = y$	$6 (w - 3) = - (2 w + 6)$

2. Solve the two equations for w

13 additional steps

$6 \cdot (w - 3) = (2 w + 6)$

Expand the parentheses:

$6 w + 6 \cdot - 3 = (2 w + 6)$

Simplify the arithmetic:

$6 w - 18 = (2 w + 6)$

Subtract from both sides:

$(6 w - 18) - 2 w = (2 w + 6) - 2 w$

Group like terms:

$(6 w - 2 w) - 18 = (2 w + 6) - 2 w$

Simplify the arithmetic:

$4 w - 18 = (2 w + 6) - 2 w$

Group like terms:

$4 w - 18 = (2 w - 2 w) + 6$

Simplify the arithmetic:

$4 w - 18 = 6$

Add to both sides:

$(4 w - 18) + 18 = 6 + 18$

Simplify the arithmetic:

$4 w = 6 + 18$

Simplify the arithmetic:

$4 w = 24$

Divide both sides by :

$\frac{(4 w)}{4} = \frac{24}{4}$

Simplify the fraction:

$w = \frac{24}{4}$

Find the greatest common factor of the numerator and denominator:

$w = \frac{(6 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$w = 6$

14 additional steps

$6 \cdot (w - 3) = - (2 w + 6)$

Expand the parentheses:

$6 w + 6 \cdot - 3 = - (2 w + 6)$

Simplify the arithmetic:

$6 w - 18 = - (2 w + 6)$

Expand the parentheses:

$6 w - 18 = - 2 w - 6$

Add to both sides:

$(6 w - 18) + 2 w = (- 2 w - 6) + 2 w$

Group like terms:

$(6 w + 2 w) - 18 = (- 2 w - 6) + 2 w$

Simplify the arithmetic:

$8 w - 18 = (- 2 w - 6) + 2 w$

Group like terms:

$8 w - 18 = (- 2 w + 2 w) - 6$

Simplify the arithmetic:

$8 w - 18 = - 6$

Add to both sides:

$(8 w - 18) + 18 = - 6 + 18$

Simplify the arithmetic:

$8 w = - 6 + 18$

Simplify the arithmetic:

$8 w = 12$

Divide both sides by :

$\frac{(8 w)}{8} = \frac{12}{8}$

Simplify the fraction:

$w = \frac{12}{8}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

$w = 6, \frac{3}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 6 | w - 3 |$
$y = | 2 w + 6 |$
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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