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Solution - Absolute value equations

Exact form:

v = - \frac{16}{5}, - 16

v=-\frac{16}{5} , -16

Mixed number form:

v = - 3 \frac{1}{5}, - 16

v=-3\frac{1}{5} , -16

Decimal form:

v = - 3.2, - 16

v=-3.2 , -16

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
$| 2 v | = | - 3 v - 16 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 v \| = \| - 3 v - 16 \|$
$x = + y$	$(2 v) = (- 3 v - 16)$
$x = - y$	$(2 v) = - (- 3 v - 16)$
$+ x = y$	$(2 v) = (- 3 v - 16)$
$- x = y$	$- (2 v) = (- 3 v - 16)$

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:

$\| x \| = \| y \|$	$\| 2 v \| = \| - 3 v - 16 \|$
$x = + y$ , $+ x = y$	$(2 v) = (- 3 v - 16)$
$x = - y$ , $- x = y$	$(2 v) = - (- 3 v - 16)$

2. Solve the two equations for v

5 additional steps

$2 v = (- 3 v - 16)$

Add to both sides:

$(2 v) + 3 v = (- 3 v - 16) + 3 v$

Simplify the arithmetic:

$5 v = (- 3 v - 16) + 3 v$

Group like terms:

$5 v = (- 3 v + 3 v) - 16$

Simplify the arithmetic:

$5 v = - 16$

Divide both sides by :

$\frac{(5 v)}{5} = \frac{- 16}{5}$

Simplify the fraction:

$v = \frac{- 16}{5}$

7 additional steps

$2 v = - (- 3 v - 16)$

Expand the parentheses:

$2 v = 3 v + 16$

Subtract from both sides:

$(2 v) - 3 v = (3 v + 16) - 3 v$

Simplify the arithmetic:

$- v = (3 v + 16) - 3 v$

Group like terms:

$- v = (3 v - 3 v) + 16$

Simplify the arithmetic:

$- v = 16$

Multiply both sides by :

$- v \cdot - 1 = 16 \cdot - 1$

Remove the one(s):

$v = 16 \cdot - 1$

Simplify the arithmetic:

$v = - 16$

3. List the solutions

$v = - \frac{16}{5}, - 16$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 v |$
$y = | - 3 v - 16 |$
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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