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Solution - Absolute value equations

Exact form:

w = 14, - \frac{10}{9}

w=14 , -\frac{10}{9}

Mixed number form:

w = 14, - 1 \frac{1}{9}

w=14 , -1\frac{1}{9}

Decimal form:

w = 14, - 1.111

w=14 , -1.111

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
$4 | w + 3 | = | 5 w - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$4 \| w + 3 \| = \| 5 w - 2 \|$
$x = + y$	$4 (w + 3) = (5 w - 2)$
$x = - y$	$4 (w + 3) = - (5 w - 2)$
$+ x = y$	$4 (w + 3) = (5 w - 2)$
$- x = y$	$4 (- (w + 3)) = (5 w - 2)$

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:

$\| x \| = \| y \|$	$4 \| w + 3 \| = \| 5 w - 2 \|$
$x = + y$ , $+ x = y$	$4 (w + 3) = (5 w - 2)$
$x = - y$ , $- x = y$	$4 (w + 3) = - (5 w - 2)$

2. Solve the two equations for w

12 additional steps

$4 \cdot (w + 3) = (5 w - 2)$

Expand the parentheses:

$4 w + 4 \cdot 3 = (5 w - 2)$

Simplify the arithmetic:

$4 w + 12 = (5 w - 2)$

Subtract from both sides:

$(4 w + 12) - 5 w = (5 w - 2) - 5 w$

Group like terms:

$(4 w - 5 w) + 12 = (5 w - 2) - 5 w$

Simplify the arithmetic:

$- w + 12 = (5 w - 2) - 5 w$

Group like terms:

$- w + 12 = (5 w - 5 w) - 2$

Simplify the arithmetic:

$- w + 12 = - 2$

Subtract from both sides:

$(- w + 12) - 12 = - 2 - 12$

Simplify the arithmetic:

$- w = - 2 - 12$

Simplify the arithmetic:

$- w = - 14$

Multiply both sides by :

$- w \cdot - 1 = - 14 \cdot - 1$

Remove the one(s):

$w = - 14 \cdot - 1$

Simplify the arithmetic:

$w = 14$

12 additional steps

$4 \cdot (w + 3) = - (5 w - 2)$

Expand the parentheses:

$4 w + 4 \cdot 3 = - (5 w - 2)$

Simplify the arithmetic:

$4 w + 12 = - (5 w - 2)$

Expand the parentheses:

$4 w + 12 = - 5 w + 2$

Add to both sides:

$(4 w + 12) + 5 w = (- 5 w + 2) + 5 w$

Group like terms:

$(4 w + 5 w) + 12 = (- 5 w + 2) + 5 w$

Simplify the arithmetic:

$9 w + 12 = (- 5 w + 2) + 5 w$

Group like terms:

$9 w + 12 = (- 5 w + 5 w) + 2$

Simplify the arithmetic:

$9 w + 12 = 2$

Subtract from both sides:

$(9 w + 12) - 12 = 2 - 12$

Simplify the arithmetic:

$9 w = 2 - 12$

Simplify the arithmetic:

$9 w = - 10$

Divide both sides by :

$\frac{(9 w)}{9} = \frac{- 10}{9}$

Simplify the fraction:

$w = \frac{- 10}{9}$

3. List the solutions

$w = 14, - \frac{10}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 4 | w + 3 |$
$y = | 5 w - 2 |$
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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