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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

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

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

Decimal form:

w = 2, - 1.5

w=2 , -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
$| w + 5 | = | 3 w + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| w + 5 \| = \| 3 w + 1 \|$
$x = + y$	$(w + 5) = (3 w + 1)$
$x = - y$	$(w + 5) = - (3 w + 1)$
$+ x = y$	$(w + 5) = (3 w + 1)$
$- x = y$	$- (w + 5) = (3 w + 1)$

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:

$\| x \| = \| y \|$	$\| w + 5 \| = \| 3 w + 1 \|$
$x = + y$ , $+ x = y$	$(w + 5) = (3 w + 1)$
$x = - y$ , $- x = y$	$(w + 5) = - (3 w + 1)$

2. Solve the two equations for w

13 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 w + 5 = (3 w + 1) - 3 w$

Group like terms:

$- 2 w + 5 = (3 w - 3 w) + 1$

Simplify the arithmetic:

$- 2 w + 5 = 1$

Subtract from both sides:

$(- 2 w + 5) - 5 = 1 - 5$

Simplify the arithmetic:

$- 2 w = 1 - 5$

Simplify the arithmetic:

$- 2 w = - 4$

Divide both sides by :

$\frac{(- 2 w)}{- 2} = \frac{- 4}{- 2}$

Cancel out the negatives:

$\frac{2 w}{2} = \frac{- 4}{- 2}$

Simplify the fraction:

$w = \frac{- 4}{- 2}$

Cancel out the negatives:

$w = \frac{4}{2}$

Find the greatest common factor of the numerator and denominator:

$w = \frac{(2 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$w = 2$

12 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$4 w + 5 = (- 3 w - 1) + 3 w$

Group like terms:

$4 w + 5 = (- 3 w + 3 w) - 1$

Simplify the arithmetic:

$4 w + 5 = - 1$

Subtract from both sides:

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

Simplify the arithmetic:

$4 w = - 1 - 5$

Simplify the arithmetic:

$4 w = - 6$

Divide both sides by :

$\frac{(4 w)}{4} = \frac{- 6}{4}$

Simplify the fraction:

$w = \frac{- 6}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | w + 5 |$
$y = | 3 w + 1 |$
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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