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Solution - Absolute value equations

Exact form:

w = \frac{1}{2}

w=\frac{1}{2}

Decimal form:

w = 0.5

w=0.5

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
$| - 2 w + 6 | = | 2 w + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 w + 6 \| = \| 2 w + 4 \|$
$x = + y$	$(- 2 w + 6) = (2 w + 4)$
$x = - y$	$(- 2 w + 6) = - (2 w + 4)$
$+ x = y$	$(- 2 w + 6) = (2 w + 4)$
$- x = y$	$- (- 2 w + 6) = (2 w + 4)$

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:

$\| x \| = \| y \|$	$\| - 2 w + 6 \| = \| 2 w + 4 \|$
$x = + y$ , $+ x = y$	$(- 2 w + 6) = (2 w + 4)$
$x = - y$ , $- x = y$	$(- 2 w + 6) = - (2 w + 4)$

2. Solve the two equations for w

13 additional steps

$(- 2 w + 6) = (2 w + 4)$

Subtract from both sides:

$(- 2 w + 6) - 2 w = (2 w + 4) - 2 w$

Group like terms:

$(- 2 w - 2 w) + 6 = (2 w + 4) - 2 w$

Simplify the arithmetic:

$- 4 w + 6 = (2 w + 4) - 2 w$

Group like terms:

$- 4 w + 6 = (2 w - 2 w) + 4$

Simplify the arithmetic:

$- 4 w + 6 = 4$

Subtract from both sides:

$(- 4 w + 6) - 6 = 4 - 6$

Simplify the arithmetic:

$- 4 w = 4 - 6$

Simplify the arithmetic:

$- 4 w = - 2$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$w = \frac{1}{2}$

6 additional steps

$(- 2 w + 6) = - (2 w + 4)$

Expand the parentheses:

$(- 2 w + 6) = - 2 w - 4$

Add to both sides:

$(- 2 w + 6) + 2 w = (- 2 w - 4) + 2 w$

Group like terms:

$(- 2 w + 2 w) + 6 = (- 2 w - 4) + 2 w$

Simplify the arithmetic:

$6 = (- 2 w - 4) + 2 w$

Group like terms:

$6 = (- 2 w + 2 w) - 4$

Simplify the arithmetic:

$6 = - 4$

The statement is false:

$6 = - 4$

The equation is false so it has no solution.

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | - 2 w + 6 |$
$y = | 2 w + 4 |$
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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