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Solution - Absolute value equations

Exact form:

v = 2, - 2

v=2 , -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
$| 3 v - 2 | = | - v + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 v - 2 \| = \| - v + 6 \|$
$x = + y$	$(3 v - 2) = (- v + 6)$
$x = - y$	$(3 v - 2) = - (- v + 6)$
$+ x = y$	$(3 v - 2) = (- v + 6)$
$- x = y$	$- (3 v - 2) = (- v + 6)$

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:

$\| x \| = \| y \|$	$\| 3 v - 2 \| = \| - v + 6 \|$
$x = + y$ , $+ x = y$	$(3 v - 2) = (- v + 6)$
$x = - y$ , $- x = y$	$(3 v - 2) = - (- v + 6)$

2. Solve the two equations for v

11 additional steps

$(3 v - 2) = (- v + 6)$

Add to both sides:

$(3 v - 2) + v = (- v + 6) + v$

Group like terms:

$(3 v + v) - 2 = (- v + 6) + v$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 v - 2 = 6$

Add to both sides:

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

Simplify the arithmetic:

$4 v = 6 + 2$

Simplify the arithmetic:

$4 v = 8$

Divide both sides by :

$\frac{(4 v)}{4} = \frac{8}{4}$

Simplify the fraction:

$v = \frac{8}{4}$

Find the greatest common factor of the numerator and denominator:

$v = \frac{(2 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$v = 2$

12 additional steps

$(3 v - 2) = - (- v + 6)$

Expand the parentheses:

$(3 v - 2) = v - 6$

Subtract from both sides:

$(3 v - 2) - v = (v - 6) - v$

Group like terms:

$(3 v - v) - 2 = (v - 6) - v$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 v - 2 = - 6$

Add to both sides:

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

Simplify the arithmetic:

$2 v = - 6 + 2$

Simplify the arithmetic:

$2 v = - 4$

Divide both sides by :

$\frac{(2 v)}{2} = \frac{- 4}{2}$

Simplify the fraction:

$v = \frac{- 4}{2}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$v = - 2$

3. List the solutions

$v = 2, - 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 v - 2 |$
$y = | - v + 6 |$
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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