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Solution - Absolute value equations

Exact form:

p = \frac{9}{2}, \frac{5}{2}

p=\frac{9}{2} , \frac{5}{2}

Mixed number form:

p = 4 \frac{1}{2}, 2 \frac{1}{2}

p=4\frac{1}{2} , 2\frac{1}{2}

Decimal form:

p = 4.5, 2.5

p=4.5 , 2.5

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

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

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:

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

2. Solve the two equations for p

13 additional steps

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

Expand the parentheses:

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

Simplify the arithmetic:

$(2 p - 3) = 4 p - 12$

Subtract from both sides:

$(2 p - 3) - 4 p = (4 p - 12) - 4 p$

Group like terms:

$(2 p - 4 p) - 3 = (4 p - 12) - 4 p$

Simplify the arithmetic:

$- 2 p - 3 = (4 p - 12) - 4 p$

Group like terms:

$- 2 p - 3 = (4 p - 4 p) - 12$

Simplify the arithmetic:

$- 2 p - 3 = - 12$

Add to both sides:

$(- 2 p - 3) + 3 = - 12 + 3$

Simplify the arithmetic:

$- 2 p = - 12 + 3$

Simplify the arithmetic:

$- 2 p = - 9$

Divide both sides by :

$\frac{(- 2 p)}{- 2} = \frac{- 9}{- 2}$

Cancel out the negatives:

$\frac{2 p}{2} = \frac{- 9}{- 2}$

Simplify the fraction:

$p = \frac{- 9}{- 2}$

Cancel out the negatives:

$p = \frac{9}{2}$

16 additional steps

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

Expand the parentheses:

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

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the arithmetic:

$(2 p - 3) = - 4 p + 12$

Add to both sides:

$(2 p - 3) + 4 p = (- 4 p + 12) + 4 p$

Group like terms:

$(2 p + 4 p) - 3 = (- 4 p + 12) + 4 p$

Simplify the arithmetic:

$6 p - 3 = (- 4 p + 12) + 4 p$

Group like terms:

$6 p - 3 = (- 4 p + 4 p) + 12$

Simplify the arithmetic:

$6 p - 3 = 12$

Add to both sides:

$(6 p - 3) + 3 = 12 + 3$

Simplify the arithmetic:

$6 p = 12 + 3$

Simplify the arithmetic:

$6 p = 15$

Divide both sides by :

$\frac{(6 p)}{6} = \frac{15}{6}$

Simplify the fraction:

$p = \frac{15}{6}$

Find the greatest common factor of the numerator and denominator:

$p = \frac{(5 \cdot 3)}{(2 \cdot 3)}$

Factor out and cancel the greatest common factor:

$p = \frac{5}{2}$

3. List the solutions

$p = \frac{9}{2}, \frac{5}{2}$
(2 solution(s))

4. Graph

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