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Solution - Absolute value equations

Exact form:

v = - \frac{14}{3}, 2

v=-\frac{14}{3} , 2

Mixed number form:

v = - 4 \frac{2}{3}, 2

v=-4\frac{2}{3} , 2

Decimal form:

v = - 4.667, 2

v=-4.667 , 2

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
$| 5 v | = | 2 v - 14 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 v \| = \| 2 v - 14 \|$
$x = + y$	$(5 v) = (2 v - 14)$
$x = - y$	$(5 v) = - (2 v - 14)$
$+ x = y$	$(5 v) = (2 v - 14)$
$- x = y$	$- (5 v) = (2 v - 14)$

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:

$\| x \| = \| y \|$	$\| 5 v \| = \| 2 v - 14 \|$
$x = + y$ , $+ x = y$	$(5 v) = (2 v - 14)$
$x = - y$ , $- x = y$	$(5 v) = - (2 v - 14)$

2. Solve the two equations for v

5 additional steps

$5 v = (2 v - 14)$

Subtract from both sides:

$(5 v) - 2 v = (2 v - 14) - 2 v$

Simplify the arithmetic:

$3 v = (2 v - 14) - 2 v$

Group like terms:

$3 v = (2 v - 2 v) - 14$

Simplify the arithmetic:

$3 v = - 14$

Divide both sides by :

$\frac{(3 v)}{3} = \frac{- 14}{3}$

Simplify the fraction:

$v = \frac{- 14}{3}$

8 additional steps

$5 v = - (2 v - 14)$

Expand the parentheses:

$5 v = - 2 v + 14$

Add to both sides:

$(5 v) + 2 v = (- 2 v + 14) + 2 v$

Simplify the arithmetic:

$7 v = (- 2 v + 14) + 2 v$

Group like terms:

$7 v = (- 2 v + 2 v) + 14$

Simplify the arithmetic:

$7 v = 14$

Divide both sides by :

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

Simplify the fraction:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$v = 2$

3. List the solutions

$v = - \frac{14}{3}, 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 v |$
$y = | 2 v - 14 |$
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 v

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 v

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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