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Solution - Absolute value equations

Exact form:

p = \frac{2}{3}, 10

p=\frac{2}{3} , 10

Decimal form:

p = 0.667, 10

p=0.667 , 10

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
$- 4 | p - 3 | = | 2 p + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$- 4 \| p - 3 \| = \| 2 p + 8 \|$
$x = + y$	$- 4 (p - 3) = (2 p + 8)$
$x = - y$	$- 4 (p - 3) = - (2 p + 8)$
$+ x = y$	$- 4 (p - 3) = (2 p + 8)$
$- x = y$	$- 4 (- (p - 3)) = (2 p + 8)$

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:

$\| x \| = \| y \|$	$- 4 \| p - 3 \| = \| 2 p + 8 \|$
$x = + y$ , $+ x = y$	$- 4 (p - 3) = (2 p + 8)$
$x = - y$ , $- x = y$	$- 4 (p - 3) = - (2 p + 8)$

2. Solve the two equations for p

15 additional steps

$- 4 \cdot (p - 3) = (2 p + 8)$

Expand the parentheses:

$- 4 p - 4 \cdot - 3 = (2 p + 8)$

Simplify the arithmetic:

$- 4 p + 12 = (2 p + 8)$

Subtract from both sides:

$(- 4 p + 12) - 2 p = (2 p + 8) - 2 p$

Group like terms:

$(- 4 p - 2 p) + 12 = (2 p + 8) - 2 p$

Simplify the arithmetic:

$- 6 p + 12 = (2 p + 8) - 2 p$

Group like terms:

$- 6 p + 12 = (2 p - 2 p) + 8$

Simplify the arithmetic:

$- 6 p + 12 = 8$

Subtract from both sides:

$(- 6 p + 12) - 12 = 8 - 12$

Simplify the arithmetic:

$- 6 p = 8 - 12$

Simplify the arithmetic:

$- 6 p = - 4$

Divide both sides by :

$\frac{(- 6 p)}{- 6} = \frac{- 4}{- 6}$

Cancel out the negatives:

$\frac{6 p}{6} = \frac{- 4}{- 6}$

Simplify the fraction:

$p = \frac{- 4}{- 6}$

Cancel out the negatives:

$p = \frac{4}{6}$

Find the greatest common factor of the numerator and denominator:

$p = \frac{(2 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$p = \frac{2}{3}$

16 additional steps

$- 4 \cdot (p - 3) = - (2 p + 8)$

Expand the parentheses:

$- 4 p - 4 \cdot - 3 = - (2 p + 8)$

Simplify the arithmetic:

$- 4 p + 12 = - (2 p + 8)$

Expand the parentheses:

$- 4 p + 12 = - 2 p - 8$

Add to both sides:

$(- 4 p + 12) + 2 p = (- 2 p - 8) + 2 p$

Group like terms:

$(- 4 p + 2 p) + 12 = (- 2 p - 8) + 2 p$

Simplify the arithmetic:

$- 2 p + 12 = (- 2 p - 8) + 2 p$

Group like terms:

$- 2 p + 12 = (- 2 p + 2 p) - 8$

Simplify the arithmetic:

$- 2 p + 12 = - 8$

Subtract from both sides:

$(- 2 p + 12) - 12 = - 8 - 12$

Simplify the arithmetic:

$- 2 p = - 8 - 12$

Simplify the arithmetic:

$- 2 p = - 20$

Divide both sides by :

$\frac{(- 2 p)}{- 2} = \frac{- 20}{- 2}$

Cancel out the negatives:

$\frac{2 p}{2} = \frac{- 20}{- 2}$

Simplify the fraction:

$p = \frac{- 20}{- 2}$

Cancel out the negatives:

$p = \frac{20}{2}$

Find the greatest common factor of the numerator and denominator:

$p = \frac{(10 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$p = 10$

3. List the solutions

$p = \frac{2}{3}, 10$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = - 4 | p - 3 |$
$y = | 2 p + 8 |$
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 p

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 p

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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