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Solution - Absolute value equations

Exact form:

w = - \frac{4}{3}, - \frac{6}{5}

w=-\frac{4}{3} , -\frac{6}{5}

Mixed number form:

w = - 1 \frac{1}{3}, - 1 \frac{1}{5}

w=-1\frac{1}{3} , -1\frac{1}{5}

Decimal form:

w = - 1.333, - 1.2

w=-1.333 , -1.2

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
$| 4 w + 5 | = | w + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 w + 5 \| = \| w + 1 \|$
$x = + y$	$(4 w + 5) = (w + 1)$
$x = - y$	$(4 w + 5) = - (w + 1)$
$+ x = y$	$(4 w + 5) = (w + 1)$
$- x = y$	$- (4 w + 5) = (w + 1)$

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:

$\| x \| = \| y \|$	$\| 4 w + 5 \| = \| w + 1 \|$
$x = + y$ , $+ x = y$	$(4 w + 5) = (w + 1)$
$x = - y$ , $- x = y$	$(4 w + 5) = - (w + 1)$

2. Solve the two equations for w

9 additional steps

$(4 w + 5) = (w + 1)$

Subtract from both sides:

$(4 w + 5) - w = (w + 1) - w$

Group like terms:

$(4 w - w) + 5 = (w + 1) - w$

Simplify the arithmetic:

$3 w + 5 = (w + 1) - w$

Group like terms:

$3 w + 5 = (w - w) + 1$

Simplify the arithmetic:

$3 w + 5 = 1$

Subtract from both sides:

$(3 w + 5) - 5 = 1 - 5$

Simplify the arithmetic:

$3 w = 1 - 5$

Simplify the arithmetic:

$3 w = - 4$

Divide both sides by :

$\frac{(3 w)}{3} = \frac{- 4}{3}$

Simplify the fraction:

$w = \frac{- 4}{3}$

10 additional steps

$(4 w + 5) = - (w + 1)$

Expand the parentheses:

$(4 w + 5) = - w - 1$

Add to both sides:

$(4 w + 5) + w = (- w - 1) + w$

Group like terms:

$(4 w + w) + 5 = (- w - 1) + w$

Simplify the arithmetic:

$5 w + 5 = (- w - 1) + w$

Group like terms:

$5 w + 5 = (- w + w) - 1$

Simplify the arithmetic:

$5 w + 5 = - 1$

Subtract from both sides:

$(5 w + 5) - 5 = - 1 - 5$

Simplify the arithmetic:

$5 w = - 1 - 5$

Simplify the arithmetic:

$5 w = - 6$

Divide both sides by :

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

Simplify the fraction:

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

3. List the solutions

$w = - \frac{4}{3}, - \frac{6}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 w + 5 |$
$y = | w + 1 |$
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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