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Solution - Absolute value equations

Exact form:

w = - \frac{1}{3}, 2

w=-\frac{1}{3} , 2

Decimal form:

w = - 0.333, 2

w=-0.333 , 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
$| - 4 w + 1 | = | 2 w + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 4 w + 1 \| = \| 2 w + 3 \|$
$x = + y$	$(- 4 w + 1) = (2 w + 3)$
$x = - y$	$(- 4 w + 1) = - (2 w + 3)$
$+ x = y$	$(- 4 w + 1) = (2 w + 3)$
$- x = y$	$- (- 4 w + 1) = (2 w + 3)$

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:

$\| x \| = \| y \|$	$\| - 4 w + 1 \| = \| 2 w + 3 \|$
$x = + y$ , $+ x = y$	$(- 4 w + 1) = (2 w + 3)$
$x = - y$ , $- x = y$	$(- 4 w + 1) = - (2 w + 3)$

2. Solve the two equations for w

13 additional steps

$(- 4 w + 1) = (2 w + 3)$

Subtract from both sides:

$(- 4 w + 1) - 2 w = (2 w + 3) - 2 w$

Group like terms:

$(- 4 w - 2 w) + 1 = (2 w + 3) - 2 w$

Simplify the arithmetic:

$- 6 w + 1 = (2 w + 3) - 2 w$

Group like terms:

$- 6 w + 1 = (2 w - 2 w) + 3$

Simplify the arithmetic:

$- 6 w + 1 = 3$

Subtract from both sides:

$(- 6 w + 1) - 1 = 3 - 1$

Simplify the arithmetic:

$- 6 w = 3 - 1$

Simplify the arithmetic:

$- 6 w = 2$

Divide both sides by :

$\frac{(- 6 w)}{- 6} = \frac{2}{- 6}$

Cancel out the negatives:

$\frac{6 w}{6} = \frac{2}{- 6}$

Simplify the fraction:

$w = \frac{2}{- 6}$

Move the negative sign from the denominator to the numerator:

$w = \frac{- 2}{6}$

Find the greatest common factor of the numerator and denominator:

$w = \frac{(- 1 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$w = \frac{- 1}{3}$

14 additional steps

$(- 4 w + 1) = - (2 w + 3)$

Expand the parentheses:

$(- 4 w + 1) = - 2 w - 3$

Add to both sides:

$(- 4 w + 1) + 2 w = (- 2 w - 3) + 2 w$

Group like terms:

$(- 4 w + 2 w) + 1 = (- 2 w - 3) + 2 w$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 w + 1 = - 3$

Subtract from both sides:

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

Simplify the arithmetic:

$- 2 w = - 3 - 1$

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:

$w = 2$

3. List the solutions

$w = - \frac{1}{3}, 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 4 w + 1 |$
$y = | 2 w + 3 |$
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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