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Solution - Absolute value equations

Exact form:

w = 1025, - 199

w=1025 , -199

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
$3 | w - 5 | = | 2 w + 1010 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| w - 5 \| = \| 2 w + 1010 \|$
$x = + y$	$3 (w - 5) = (2 w + 1010)$
$x = - y$	$3 (w - 5) = - (2 w + 1010)$
$+ x = y$	$3 (w - 5) = (2 w + 1010)$
$- x = y$	$3 (- (w - 5)) = (2 w + 1010)$

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:

$\| x \| = \| y \|$	$3 \| w - 5 \| = \| 2 w + 1010 \|$
$x = + y$ , $+ x = y$	$3 (w - 5) = (2 w + 1010)$
$x = - y$ , $- x = y$	$3 (w - 5) = - (2 w + 1010)$

2. Solve the two equations for w

9 additional steps

$3 \cdot (w - 5) = (2 w + 1010)$

Expand the parentheses:

$3 w + 3 \cdot - 5 = (2 w + 1010)$

Simplify the arithmetic:

$3 w - 15 = (2 w + 1010)$

Subtract from both sides:

$(3 w - 15) - 2 w = (2 w + 1010) - 2 w$

Group like terms:

$(3 w - 2 w) - 15 = (2 w + 1010) - 2 w$

Simplify the arithmetic:

$w - 15 = (2 w + 1010) - 2 w$

Group like terms:

$w - 15 = (2 w - 2 w) + 1010$

Simplify the arithmetic:

$w - 15 = 1010$

Add to both sides:

$(w - 15) + 15 = 1010 + 15$

Simplify the arithmetic:

$w = 1010 + 15$

Simplify the arithmetic:

$w = 1025$

14 additional steps

$3 \cdot (w - 5) = - (2 w + 1010)$

Expand the parentheses:

$3 w + 3 \cdot - 5 = - (2 w + 1010)$

Simplify the arithmetic:

$3 w - 15 = - (2 w + 1010)$

Expand the parentheses:

$3 w - 15 = - 2 w - 1010$

Add to both sides:

$(3 w - 15) + 2 w = (- 2 w - 1010) + 2 w$

Group like terms:

$(3 w + 2 w) - 15 = (- 2 w - 1010) + 2 w$

Simplify the arithmetic:

$5 w - 15 = (- 2 w - 1010) + 2 w$

Group like terms:

$5 w - 15 = (- 2 w + 2 w) - 1010$

Simplify the arithmetic:

$5 w - 15 = - 1010$

Add to both sides:

$(5 w - 15) + 15 = - 1010 + 15$

Simplify the arithmetic:

$5 w = - 1010 + 15$

Simplify the arithmetic:

$5 w = - 995$

Divide both sides by :

$\frac{(5 w)}{5} = \frac{- 995}{5}$

Simplify the fraction:

$w = \frac{- 995}{5}$

Find the greatest common factor of the numerator and denominator:

$w = \frac{(- 199 \cdot 5)}{(1 \cdot 5)}$

Factor out and cancel the greatest common factor:

$w = - 199$

3. List the solutions

$w = 1025, - 199$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | w - 5 |$
$y = | 2 w + 1010 |$
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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