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Solution - Absolute value equations

Exact form:

v = \frac{1}{2}

v=\frac{1}{2}

Decimal form:

v = 0.5

v=0.5

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

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

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:

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

2. Solve the two equations for v

13 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 4 v - 4 = - 6$

Add to both sides:

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

Simplify the arithmetic:

$- 4 v = - 6 + 4$

Simplify the arithmetic:

$- 4 v = - 2$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$v = \frac{1}{2}$

6 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 4 = 6$

The statement is false:

$- 4 = 6$

The equation is false so it has no solution.

3. List the solutions

$v = \frac{1}{2}$
(1 solution(s))

4. Graph

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