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Solution - Absolute value equations

Exact form:

v = 72, \frac{99}{2}

v=72 , \frac{99}{2}

Mixed number form:

v = 72, 49 \frac{1}{2}

v=72 , 49\frac{1}{2}

Decimal form:

v = 72, 49.5

v=72 , 49.5

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| - v + 27 | - 3 | - v + 57 | = 0$

Add $3 | - v + 57 |$ to both sides of the equation:

$| - v + 27 | - 3 | - v + 57 | + 3 | - v + 57 | = 3 | - v + 57 |$

Simplify the arithmetic

$| - v + 27 | = 3 | - v + 57 |$

2. 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
$| - v + 27 | = 3 | - v + 57 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - v + 27 \| = 3 \| - v + 57 \|$
$x = + y$	$(- v + 27) = 3 (- v + 57)$
$x = - y$	$(- v + 27) = 3 (- (- v + 57))$
$+ x = y$	$(- v + 27) = 3 (- v + 57)$
$- x = y$	$- (- v + 27) = 3 (- v + 57)$

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:

$\| x \| = \| y \|$	$\| - v + 27 \| = 3 \| - v + 57 \|$
$x = + y$ , $+ x = y$	$(- v + 27) = 3 (- v + 57)$
$x = - y$ , $- x = y$	$(- v + 27) = 3 (- (- v + 57))$

3. Solve the two equations for v

15 additional steps

$(- v + 27) = 3 \cdot (- v + 57)$

Expand the parentheses:

$(- v + 27) = 3 \cdot - v + 3 \cdot 57$

Group like terms:

$(- v + 27) = (3 \cdot - 1) v + 3 \cdot 57$

Multiply the coefficients:

$(- v + 27) = - 3 v + 3 \cdot 57$

Simplify the arithmetic:

$(- v + 27) = - 3 v + 171$

Add to both sides:

$(- v + 27) + 3 v = (- 3 v + 171) + 3 v$

Group like terms:

$(- v + 3 v) + 27 = (- 3 v + 171) + 3 v$

Simplify the arithmetic:

$2 v + 27 = (- 3 v + 171) + 3 v$

Group like terms:

$2 v + 27 = (- 3 v + 3 v) + 171$

Simplify the arithmetic:

$2 v + 27 = 171$

Subtract from both sides:

$(2 v + 27) - 27 = 171 - 27$

Simplify the arithmetic:

$2 v = 171 - 27$

Simplify the arithmetic:

$2 v = 144$

Divide both sides by :

$\frac{(2 v)}{2} = \frac{144}{2}$

Simplify the fraction:

$v = \frac{144}{2}$

Find the greatest common factor of the numerator and denominator:

$v = \frac{(72 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$v = 72$

16 additional steps

$(- v + 27) = 3 \cdot (- (- v + 57))$

Expand the parentheses:

$(- v + 27) = 3 \cdot (v - 57)$

$(- v + 27) = 3 v + 3 \cdot - 57$

Simplify the arithmetic:

$(- v + 27) = 3 v - 171$

Subtract from both sides:

$(- v + 27) - 3 v = (3 v - 171) - 3 v$

Group like terms:

$(- v - 3 v) + 27 = (3 v - 171) - 3 v$

Simplify the arithmetic:

$- 4 v + 27 = (3 v - 171) - 3 v$

Group like terms:

$- 4 v + 27 = (3 v - 3 v) - 171$

Simplify the arithmetic:

$- 4 v + 27 = - 171$

Subtract from both sides:

$(- 4 v + 27) - 27 = - 171 - 27$

Simplify the arithmetic:

$- 4 v = - 171 - 27$

Simplify the arithmetic:

$- 4 v = - 198$

Divide both sides by :

$\frac{(- 4 v)}{- 4} = \frac{- 198}{- 4}$

Cancel out the negatives:

$\frac{4 v}{4} = \frac{- 198}{- 4}$

Simplify the fraction:

$v = \frac{- 198}{- 4}$

Cancel out the negatives:

$v = \frac{198}{4}$

Find the greatest common factor of the numerator and denominator:

$v = \frac{(99 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$v = \frac{99}{2}$

4. List the solutions

$v = 72, \frac{99}{2}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - v + 27 |$
$y = 3 | - v + 57 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for v

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for v

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved