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Solution - Absolute value equations

Exact form:

v = - \frac{9}{7}, 3

v=-\frac{9}{7} , 3

Mixed number form:

v = - 1 \frac{2}{7}, 3

v=-1\frac{2}{7} , 3

Decimal form:

v = - 1.286, 3

v=-1.286 , 3

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
$| - 3 v - 6 | = | 4 v + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 v - 6 \| = \| 4 v + 3 \|$
$x = + y$	$(- 3 v - 6) = (4 v + 3)$
$x = - y$	$(- 3 v - 6) = - (4 v + 3)$
$+ x = y$	$(- 3 v - 6) = (4 v + 3)$
$- x = y$	$- (- 3 v - 6) = (4 v + 3)$

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:

$\| x \| = \| y \|$	$\| - 3 v - 6 \| = \| 4 v + 3 \|$
$x = + y$ , $+ x = y$	$(- 3 v - 6) = (4 v + 3)$
$x = - y$ , $- x = y$	$(- 3 v - 6) = - (4 v + 3)$

2. Solve the two equations for v

11 additional steps

$(- 3 v - 6) = (4 v + 3)$

Subtract from both sides:

$(- 3 v - 6) - 4 v = (4 v + 3) - 4 v$

Group like terms:

$(- 3 v - 4 v) - 6 = (4 v + 3) - 4 v$

Simplify the arithmetic:

$- 7 v - 6 = (4 v + 3) - 4 v$

Group like terms:

$- 7 v - 6 = (4 v - 4 v) + 3$

Simplify the arithmetic:

$- 7 v - 6 = 3$

Add to both sides:

$(- 7 v - 6) + 6 = 3 + 6$

Simplify the arithmetic:

$- 7 v = 3 + 6$

Simplify the arithmetic:

$- 7 v = 9$

Divide both sides by :

$\frac{(- 7 v)}{- 7} = \frac{9}{- 7}$

Cancel out the negatives:

$\frac{7 v}{7} = \frac{9}{- 7}$

Simplify the fraction:

$v = \frac{9}{- 7}$

Move the negative sign from the denominator to the numerator:

$v = \frac{- 9}{7}$

8 additional steps

$(- 3 v - 6) = - (4 v + 3)$

Expand the parentheses:

$(- 3 v - 6) = - 4 v - 3$

Add to both sides:

$(- 3 v - 6) + 4 v = (- 4 v - 3) + 4 v$

Group like terms:

$(- 3 v + 4 v) - 6 = (- 4 v - 3) + 4 v$

Simplify the arithmetic:

$v - 6 = (- 4 v - 3) + 4 v$

Group like terms:

$v - 6 = (- 4 v + 4 v) - 3$

Simplify the arithmetic:

$v - 6 = - 3$

Add to both sides:

$(v - 6) + 6 = - 3 + 6$

Simplify the arithmetic:

$v = - 3 + 6$

Simplify the arithmetic:

$v = 3$

3. List the solutions

$v = - \frac{9}{7}, 3$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 3 v - 6 |$
$y = | 4 v + 3 |$
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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