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Solution - Absolute value equations

Exact form:

w = - \frac{1}{8}

w=-\frac{1}{8}

Decimal form:

w = - 0.125

w=-0.125

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

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

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:

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

2. Solve the two equations for w

11 additional steps

$4 \cdot (4 w - 1) = 4 \cdot (4 w + 2)$

Expand the parentheses:

$4 \cdot 4 w + 4 \cdot - 1 = 4 \cdot (4 w + 2)$

Multiply the coefficients:

$16 w + 4 \cdot - 1 = 4 \cdot (4 w + 2)$

Simplify the arithmetic:

$16 w - 4 = 4 \cdot (4 w + 2)$

Expand the parentheses:

$16 w - 4 = 4 \cdot 4 w + 4 \cdot 2$

Multiply the coefficients:

$16 w - 4 = 16 w + 4 \cdot 2$

Simplify the arithmetic:

$16 w - 4 = 16 w + 8$

Subtract from both sides:

$(16 w - 4) - 16 w = (16 w + 8) - 16 w$

Group like terms:

$(16 w - 16 w) - 4 = (16 w + 8) - 16 w$

Simplify the arithmetic:

$- 4 = (16 w + 8) - 16 w$

Group like terms:

$- 4 = (16 w - 16 w) + 8$

Simplify the arithmetic:

$- 4 = 8$

The statement is false:

$- 4 = 8$

The equation is false so it has no solution.

18 additional steps

$4 \cdot (4 w - 1) = 4 \cdot (- (4 w + 2))$

Expand the parentheses:

$4 \cdot 4 w + 4 \cdot - 1 = 4 \cdot (- (4 w + 2))$

Multiply the coefficients:

$16 w + 4 \cdot - 1 = 4 \cdot (- (4 w + 2))$

Simplify the arithmetic:

$16 w - 4 = 4 \cdot (- (4 w + 2))$

Expand the parentheses:

$16 w - 4 = 4 \cdot (- 4 w - 2)$

Expand the parentheses:

$16 w - 4 = 4 \cdot - 4 w + 4 \cdot - 2$

Multiply the coefficients:

$16 w - 4 = - 16 w + 4 \cdot - 2$

Simplify the arithmetic:

$16 w - 4 = - 16 w - 8$

Add to both sides:

$(16 w - 4) + 16 w = (- 16 w - 8) + 16 w$

Group like terms:

$(16 w + 16 w) - 4 = (- 16 w - 8) + 16 w$

Simplify the arithmetic:

$32 w - 4 = (- 16 w - 8) + 16 w$

Group like terms:

$32 w - 4 = (- 16 w + 16 w) - 8$

Simplify the arithmetic:

$32 w - 4 = - 8$

Add to both sides:

$(32 w - 4) + 4 = - 8 + 4$

Simplify the arithmetic:

$32 w = - 8 + 4$

Simplify the arithmetic:

$32 w = - 4$

Divide both sides by :

$\frac{(32 w)}{32} = \frac{- 4}{32}$

Simplify the fraction:

$w = \frac{- 4}{32}$

Find the greatest common factor of the numerator and denominator:

$w = \frac{(- 1 \cdot 4)}{(8 \cdot 4)}$

Factor out and cancel the greatest common factor:

$w = \frac{- 1}{8}$

3. Graph

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

Why learn this

Terms and topics

Related links

Latest Related Drills Solved