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Solution - Absolute value equations

Exact form:

v = \frac{6}{5}, - 8

v=\frac{6}{5} , -8

Mixed number form:

v = 1 \frac{1}{5}, - 8

v=1\frac{1}{5} , -8

Decimal form:

v = 1.2, - 8

v=1.2 , -8

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
$| - 3 v - 1 | = | 2 v - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 v - 1 \| = \| 2 v - 7 \|$
$x = + y$	$(- 3 v - 1) = (2 v - 7)$
$x = - y$	$(- 3 v - 1) = - (2 v - 7)$
$+ x = y$	$(- 3 v - 1) = (2 v - 7)$
$- x = y$	$- (- 3 v - 1) = (2 v - 7)$

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:

$\| x \| = \| y \|$	$\| - 3 v - 1 \| = \| 2 v - 7 \|$
$x = + y$ , $+ x = y$	$(- 3 v - 1) = (2 v - 7)$
$x = - y$ , $- x = y$	$(- 3 v - 1) = - (2 v - 7)$

2. Solve the two equations for v

11 additional steps

$(- 3 v - 1) = (2 v - 7)$

Subtract from both sides:

$(- 3 v - 1) - 2 v = (2 v - 7) - 2 v$

Group like terms:

$(- 3 v - 2 v) - 1 = (2 v - 7) - 2 v$

Simplify the arithmetic:

$- 5 v - 1 = (2 v - 7) - 2 v$

Group like terms:

$- 5 v - 1 = (2 v - 2 v) - 7$

Simplify the arithmetic:

$- 5 v - 1 = - 7$

Add to both sides:

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

Simplify the arithmetic:

$- 5 v = - 7 + 1$

Simplify the arithmetic:

$- 5 v = - 6$

Divide both sides by :

$\frac{(- 5 v)}{- 5} = \frac{- 6}{- 5}$

Cancel out the negatives:

$\frac{5 v}{5} = \frac{- 6}{- 5}$

Simplify the fraction:

$v = \frac{- 6}{- 5}$

Cancel out the negatives:

$v = \frac{6}{5}$

11 additional steps

$(- 3 v - 1) = - (2 v - 7)$

Expand the parentheses:

$(- 3 v - 1) = - 2 v + 7$

Add to both sides:

$(- 3 v - 1) + 2 v = (- 2 v + 7) + 2 v$

Group like terms:

$(- 3 v + 2 v) - 1 = (- 2 v + 7) + 2 v$

Simplify the arithmetic:

$- v - 1 = (- 2 v + 7) + 2 v$

Group like terms:

$- v - 1 = (- 2 v + 2 v) + 7$

Simplify the arithmetic:

$- v - 1 = 7$

Add to both sides:

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

Simplify the arithmetic:

$- v = 7 + 1$

Simplify the arithmetic:

$- v = 8$

Multiply both sides by :

$- v \cdot - 1 = 8 \cdot - 1$

Remove the one(s):

$v = 8 \cdot - 1$

Simplify the arithmetic:

$v = - 8$

3. List the solutions

$v = \frac{6}{5}, - 8$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 3 v - 1 |$
$y = | 2 v - 7 |$
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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