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Solution - Absolute value equations

Exact form:

w = \frac{8}{7}, 8

w=\frac{8}{7} , 8

Mixed number form:

w = 1 \frac{1}{7}, 8

w=1\frac{1}{7} , 8

Decimal form:

w = 1.143, 8

w=1.143 , 8

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
$| - 7 w + 8 | = | 7 w - 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 7 w + 8 \| = \| 7 w - 8 \|$
$x = + y$	$(- 7 w + 8) = (7 w - 8)$
$x = - y$	$(- 7 w + 8) = - (7 w - 8)$
$+ x = y$	$(- 7 w + 8) = (7 w - 8)$
$- x = y$	$- (- 7 w + 8) = (7 w - 8)$

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:

$\| x \| = \| y \|$	$\| - 7 w + 8 \| = \| 7 w - 8 \|$
$x = + y$ , $+ x = y$	$(- 7 w + 8) = (7 w - 8)$
$x = - y$ , $- x = y$	$(- 7 w + 8) = - (7 w - 8)$

2. Solve the two equations for w

13 additional steps

$(- 7 w + 8) = (7 w - 8)$

Subtract from both sides:

$(- 7 w + 8) - 7 w = (7 w - 8) - 7 w$

Group like terms:

$(- 7 w - 7 w) + 8 = (7 w - 8) - 7 w$

Simplify the arithmetic:

$- 14 w + 8 = (7 w - 8) - 7 w$

Group like terms:

$- 14 w + 8 = (7 w - 7 w) - 8$

Simplify the arithmetic:

$- 14 w + 8 = - 8$

Subtract from both sides:

$(- 14 w + 8) - 8 = - 8 - 8$

Simplify the arithmetic:

$- 14 w = - 8 - 8$

Simplify the arithmetic:

$- 14 w = - 16$

Divide both sides by :

$\frac{(- 14 w)}{- 14} = \frac{- 16}{- 14}$

Cancel out the negatives:

$\frac{14 w}{14} = \frac{- 16}{- 14}$

Simplify the fraction:

$w = \frac{- 16}{- 14}$

Cancel out the negatives:

$w = \frac{16}{14}$

Find the greatest common factor of the numerator and denominator:

$w = \frac{(8 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$w = \frac{8}{7}$

5 additional steps

$(- 7 w + 8) = - (7 w - 8)$

Expand the parentheses:

$(- 7 w + 8) = - 7 w + 8$

Add to both sides:

$(- 7 w + 8) + 7 w = (- 7 w + 8) + 7 w$

Group like terms:

$(- 7 w + 7 w) + 8 = (- 7 w + 8) + 7 w$

Simplify the arithmetic:

$8 = (- 7 w + 8) + 7 w$

Group like terms:

$8 = (- 7 w + 7 w) + 8$

Simplify the arithmetic:

$8 = 8$

3. List the solutions

$w = \frac{8}{7}, 8$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 7 w + 8 |$
$y = | 7 w - 8 |$
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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