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Solution - Absolute value equations

Exact form:

w = 2, - \frac{6}{11}

w=2 , -\frac{6}{11}

Decimal form:

w = 2, - 0.545

w=2 , -0.545

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
$| 9 w - 4 | = | 2 w + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 w - 4 \| = \| 2 w + 10 \|$
$x = + y$	$(9 w - 4) = (2 w + 10)$
$x = - y$	$(9 w - 4) = - (2 w + 10)$
$+ x = y$	$(9 w - 4) = (2 w + 10)$
$- x = y$	$- (9 w - 4) = (2 w + 10)$

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:

$\| x \| = \| y \|$	$\| 9 w - 4 \| = \| 2 w + 10 \|$
$x = + y$ , $+ x = y$	$(9 w - 4) = (2 w + 10)$
$x = - y$ , $- x = y$	$(9 w - 4) = - (2 w + 10)$

2. Solve the two equations for w

11 additional steps

$(9 w - 4) = (2 w + 10)$

Subtract from both sides:

$(9 w - 4) - 2 w = (2 w + 10) - 2 w$

Group like terms:

$(9 w - 2 w) - 4 = (2 w + 10) - 2 w$

Simplify the arithmetic:

$7 w - 4 = (2 w + 10) - 2 w$

Group like terms:

$7 w - 4 = (2 w - 2 w) + 10$

Simplify the arithmetic:

$7 w - 4 = 10$

Add to both sides:

$(7 w - 4) + 4 = 10 + 4$

Simplify the arithmetic:

$7 w = 10 + 4$

Simplify the arithmetic:

$7 w = 14$

Divide both sides by :

$\frac{(7 w)}{7} = \frac{14}{7}$

Simplify the fraction:

$w = \frac{14}{7}$

Find the greatest common factor of the numerator and denominator:

$w = \frac{(2 \cdot 7)}{(1 \cdot 7)}$

Factor out and cancel the greatest common factor:

$w = 2$

10 additional steps

$(9 w - 4) = - (2 w + 10)$

Expand the parentheses:

$(9 w - 4) = - 2 w - 10$

Add to both sides:

$(9 w - 4) + 2 w = (- 2 w - 10) + 2 w$

Group like terms:

$(9 w + 2 w) - 4 = (- 2 w - 10) + 2 w$

Simplify the arithmetic:

$11 w - 4 = (- 2 w - 10) + 2 w$

Group like terms:

$11 w - 4 = (- 2 w + 2 w) - 10$

Simplify the arithmetic:

$11 w - 4 = - 10$

Add to both sides:

$(11 w - 4) + 4 = - 10 + 4$

Simplify the arithmetic:

$11 w = - 10 + 4$

Simplify the arithmetic:

$11 w = - 6$

Divide both sides by :

$\frac{(11 w)}{11} = \frac{- 6}{11}$

Simplify the fraction:

$w = \frac{- 6}{11}$

3. List the solutions

$w = 2, - \frac{6}{11}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 w - 4 |$
$y = | 2 w + 10 |$
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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