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Solution - Absolute value equations

Exact form:

w = - 8, \frac{2}{3}

w=-8 , \frac{2}{3}

Decimal form:

w = - 8, 0.667

w=-8 , 0.667

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
$| 4 w - 7 | = | 5 w + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 w - 7 \| = \| 5 w + 1 \|$
$x = + y$	$(4 w - 7) = (5 w + 1)$
$x = - y$	$(4 w - 7) = - (5 w + 1)$
$+ x = y$	$(4 w - 7) = (5 w + 1)$
$- x = y$	$- (4 w - 7) = (5 w + 1)$

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:

$\| x \| = \| y \|$	$\| 4 w - 7 \| = \| 5 w + 1 \|$
$x = + y$ , $+ x = y$	$(4 w - 7) = (5 w + 1)$
$x = - y$ , $- x = y$	$(4 w - 7) = - (5 w + 1)$

2. Solve the two equations for w

10 additional steps

$(4 w - 7) = (5 w + 1)$

Subtract from both sides:

$(4 w - 7) - 5 w = (5 w + 1) - 5 w$

Group like terms:

$(4 w - 5 w) - 7 = (5 w + 1) - 5 w$

Simplify the arithmetic:

$- w - 7 = (5 w + 1) - 5 w$

Group like terms:

$- w - 7 = (5 w - 5 w) + 1$

Simplify the arithmetic:

$- w - 7 = 1$

Add to both sides:

$(- w - 7) + 7 = 1 + 7$

Simplify the arithmetic:

$- w = 1 + 7$

Simplify the arithmetic:

$- w = 8$

Multiply both sides by :

$- w \cdot - 1 = 8 \cdot - 1$

Remove the one(s):

$w = 8 \cdot - 1$

Simplify the arithmetic:

$w = - 8$

12 additional steps

$(4 w - 7) = - (5 w + 1)$

Expand the parentheses:

$(4 w - 7) = - 5 w - 1$

Add to both sides:

$(4 w - 7) + 5 w = (- 5 w - 1) + 5 w$

Group like terms:

$(4 w + 5 w) - 7 = (- 5 w - 1) + 5 w$

Simplify the arithmetic:

$9 w - 7 = (- 5 w - 1) + 5 w$

Group like terms:

$9 w - 7 = (- 5 w + 5 w) - 1$

Simplify the arithmetic:

$9 w - 7 = - 1$

Add to both sides:

$(9 w - 7) + 7 = - 1 + 7$

Simplify the arithmetic:

$9 w = - 1 + 7$

Simplify the arithmetic:

$9 w = 6$

Divide both sides by :

$\frac{(9 w)}{9} = \frac{6}{9}$

Simplify the fraction:

$w = \frac{6}{9}$

Find the greatest common factor of the numerator and denominator:

$w = \frac{(2 \cdot 3)}{(3 \cdot 3)}$

Factor out and cancel the greatest common factor:

$w = \frac{2}{3}$

3. List the solutions

$w = - 8, \frac{2}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 w - 7 |$
$y = | 5 w + 1 |$
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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