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Solution - Absolute value equations

Exact form:

w = - \frac{5}{2}, \frac{5}{6}

w=-\frac{5}{2} , \frac{5}{6}

Mixed number form:

w = - 2 \frac{1}{2}, \frac{5}{6}

w=-2\frac{1}{2} , \frac{5}{6}

Decimal form:

w = - 2.5, 0.833

w=-2.5 , 0.833

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
$| 4 w | = | 2 w - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 w \| = \| 2 w - 5 \|$
$x = + y$	$(4 w) = (2 w - 5)$
$x = - y$	$(4 w) = - (2 w - 5)$
$+ x = y$	$(4 w) = (2 w - 5)$
$- x = y$	$- (4 w) = (2 w - 5)$

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:

$\| x \| = \| y \|$	$\| 4 w \| = \| 2 w - 5 \|$
$x = + y$ , $+ x = y$	$(4 w) = (2 w - 5)$
$x = - y$ , $- x = y$	$(4 w) = - (2 w - 5)$

2. Solve the two equations for w

5 additional steps

$4 w = (2 w - 5)$

Subtract from both sides:

$(4 w) - 2 w = (2 w - 5) - 2 w$

Simplify the arithmetic:

$2 w = (2 w - 5) - 2 w$

Group like terms:

$2 w = (2 w - 2 w) - 5$

Simplify the arithmetic:

$2 w = - 5$

Divide both sides by :

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

Simplify the fraction:

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

6 additional steps

$4 w = - (2 w - 5)$

Expand the parentheses:

$4 w = - 2 w + 5$

Add to both sides:

$(4 w) + 2 w = (- 2 w + 5) + 2 w$

Simplify the arithmetic:

$6 w = (- 2 w + 5) + 2 w$

Group like terms:

$6 w = (- 2 w + 2 w) + 5$

Simplify the arithmetic:

$6 w = 5$

Divide both sides by :

$\frac{(6 w)}{6} = \frac{5}{6}$

Simplify the fraction:

$w = \frac{5}{6}$

3. List the solutions

$w = - \frac{5}{2}, \frac{5}{6}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 w |$
$y = | 2 w - 5 |$
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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