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Solution - Absolute value equations

Exact form:

p = \frac{29}{2}, \frac{35}{6}

p=\frac{29}{2} , \frac{35}{6}

Mixed number form:

p = 14 \frac{1}{2}, 5 \frac{5}{6}

p=14\frac{1}{2} , 5\frac{5}{6}

Decimal form:

p = 14.5, 5.833

p=14.5 , 5.833

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

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

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:

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

2. Solve the two equations for p

13 additional steps

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

Expand the parentheses:

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

Simplify the arithmetic:

$(2 p - 3) = 4 p - 32$

Subtract from both sides:

$(2 p - 3) - 4 p = (4 p - 32) - 4 p$

Group like terms:

$(2 p - 4 p) - 3 = (4 p - 32) - 4 p$

Simplify the arithmetic:

$- 2 p - 3 = (4 p - 32) - 4 p$

Group like terms:

$- 2 p - 3 = (4 p - 4 p) - 32$

Simplify the arithmetic:

$- 2 p - 3 = - 32$

Add to both sides:

$(- 2 p - 3) + 3 = - 32 + 3$

Simplify the arithmetic:

$- 2 p = - 32 + 3$

Simplify the arithmetic:

$- 2 p = - 29$

Divide both sides by :

$\frac{(- 2 p)}{- 2} = \frac{- 29}{- 2}$

Cancel out the negatives:

$\frac{2 p}{2} = \frac{- 29}{- 2}$

Simplify the fraction:

$p = \frac{- 29}{- 2}$

Cancel out the negatives:

$p = \frac{29}{2}$

14 additional steps

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

Expand the parentheses:

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

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

Group like terms:

$(2 p - 3) = (4 \cdot - 1) p + 4 \cdot 8$

Multiply the coefficients:

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

Simplify the arithmetic:

$(2 p - 3) = - 4 p + 32$

Add to both sides:

$(2 p - 3) + 4 p = (- 4 p + 32) + 4 p$

Group like terms:

$(2 p + 4 p) - 3 = (- 4 p + 32) + 4 p$

Simplify the arithmetic:

$6 p - 3 = (- 4 p + 32) + 4 p$

Group like terms:

$6 p - 3 = (- 4 p + 4 p) + 32$

Simplify the arithmetic:

$6 p - 3 = 32$

Add to both sides:

$(6 p - 3) + 3 = 32 + 3$

Simplify the arithmetic:

$6 p = 32 + 3$

Simplify the arithmetic:

$6 p = 35$

Divide both sides by :

$\frac{(6 p)}{6} = \frac{35}{6}$

Simplify the fraction:

$p = \frac{35}{6}$

3. List the solutions

$p = \frac{29}{2}, \frac{35}{6}$
(2 solution(s))

4. Graph

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