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Solution - Absolute value equations

Exact form:

w = \frac{1}{9}

w=\frac{1}{9}

Decimal form:

w = 0.111

w=0.111

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
$| w | = | w - \frac{2}{9} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| w \| = \| w - \frac{2}{9} \|$
$x = + y$	$(w) = (w - \frac{2}{9})$
$x = - y$	$(w) = - (w - \frac{2}{9})$
$+ x = y$	$(w) = (w - \frac{2}{9})$
$- x = y$	$- (w) = (w - \frac{2}{9})$

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:

$\| x \| = \| y \|$	$\| w \| = \| w - \frac{2}{9} \|$
$x = + y$ , $+ x = y$	$(w) = (w - \frac{2}{9})$
$x = - y$ , $- x = y$	$(w) = - (w - \frac{2}{9})$

2. Solve the two equations for w

4 additional steps

$w = (w + \frac{- 2}{9})$

Subtract from both sides:

$w - w = (w + \frac{- 2}{9}) - w$

Simplify the arithmetic:

$0 = (w + \frac{- 2}{9}) - w$

Group like terms:

$0 = (w - w) + \frac{- 2}{9}$

Simplify the arithmetic:

$0 = \frac{- 2}{9}$

The statement is false:

$0 = \frac{- 2}{9}$

The equation is false so it has no solution.

8 additional steps

$w = - (w + \frac{- 2}{9})$

Expand the parentheses:

$w = - w + \frac{2}{9}$

Add to both sides:

$w + w = (- w + \frac{2}{9}) + w$

Simplify the arithmetic:

$2 w = (- w + \frac{2}{9}) + w$

Group like terms:

$2 w = (- w + w) + \frac{2}{9}$

Simplify the arithmetic:

$2 w = \frac{2}{9}$

Divide both sides by :

$\frac{(2 w)}{2} = \frac{(\frac{2}{9})}{2}$

Simplify the fraction:

$w = \frac{(\frac{2}{9})}{2}$

Simplify the arithmetic:

$w = \frac{2}{(9 \cdot 2)}$

$w = \frac{1}{9}$

3. Graph

Each line represents the function of one side of the equation:
$y = | w |$
$y = | w - \frac{2}{9} |$
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