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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

v = - 0.333, - 5

v=-0.333 , -5

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
$| 4 v + 6 | = | - 2 v + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 v + 6 \| = \| - 2 v + 4 \|$
$x = + y$	$(4 v + 6) = (- 2 v + 4)$
$x = - y$	$(4 v + 6) = - (- 2 v + 4)$
$+ x = y$	$(4 v + 6) = (- 2 v + 4)$
$- x = y$	$- (4 v + 6) = (- 2 v + 4)$

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:

$\| x \| = \| y \|$	$\| 4 v + 6 \| = \| - 2 v + 4 \|$
$x = + y$ , $+ x = y$	$(4 v + 6) = (- 2 v + 4)$
$x = - y$ , $- x = y$	$(4 v + 6) = - (- 2 v + 4)$

2. Solve the two equations for v

11 additional steps

$(4 v + 6) = (- 2 v + 4)$

Add to both sides:

$(4 v + 6) + 2 v = (- 2 v + 4) + 2 v$

Group like terms:

$(4 v + 2 v) + 6 = (- 2 v + 4) + 2 v$

Simplify the arithmetic:

$6 v + 6 = (- 2 v + 4) + 2 v$

Group like terms:

$6 v + 6 = (- 2 v + 2 v) + 4$

Simplify the arithmetic:

$6 v + 6 = 4$

Subtract from both sides:

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

Simplify the arithmetic:

$6 v = 4 - 6$

Simplify the arithmetic:

$6 v = - 2$

Divide both sides by :

$\frac{(6 v)}{6} = \frac{- 2}{6}$

Simplify the fraction:

$v = \frac{- 2}{6}$

Find the greatest common factor of the numerator and denominator:

$v = \frac{(- 1 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$v = \frac{- 1}{3}$

12 additional steps

$(4 v + 6) = - (- 2 v + 4)$

Expand the parentheses:

$(4 v + 6) = 2 v - 4$

Subtract from both sides:

$(4 v + 6) - 2 v = (2 v - 4) - 2 v$

Group like terms:

$(4 v - 2 v) + 6 = (2 v - 4) - 2 v$

Simplify the arithmetic:

$2 v + 6 = (2 v - 4) - 2 v$

Group like terms:

$2 v + 6 = (2 v - 2 v) - 4$

Simplify the arithmetic:

$2 v + 6 = - 4$

Subtract from both sides:

$(2 v + 6) - 6 = - 4 - 6$

Simplify the arithmetic:

$2 v = - 4 - 6$

Simplify the arithmetic:

$2 v = - 10$

Divide both sides by :

$\frac{(2 v)}{2} = \frac{- 10}{2}$

Simplify the fraction:

$v = \frac{- 10}{2}$

Find the greatest common factor of the numerator and denominator:

$v = \frac{(- 5 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$v = - 5$

3. List the solutions

$v = - \frac{1}{3}, - 5$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 v + 6 |$
$y = | - 2 v + 4 |$
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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