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Solution - Absolute value equations

Exact form:

p = \frac{10}{7}, \frac{2}{9}

p=\frac{10}{7} , \frac{2}{9}

Mixed number form:

p = 1 \frac{3}{7}, \frac{2}{9}

p=1\frac{3}{7} , \frac{2}{9}

Decimal form:

p = 1.429, 0.222

p=1.429 , 0.222

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

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

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:

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

2. Solve the two equations for p

14 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$16 p + 4 \cdot - 3 = (2 p + 8)$

Simplify the arithmetic:

$16 p - 12 = (2 p + 8)$

Subtract from both sides:

$(16 p - 12) - 2 p = (2 p + 8) - 2 p$

Group like terms:

$(16 p - 2 p) - 12 = (2 p + 8) - 2 p$

Simplify the arithmetic:

$14 p - 12 = (2 p + 8) - 2 p$

Group like terms:

$14 p - 12 = (2 p - 2 p) + 8$

Simplify the arithmetic:

$14 p - 12 = 8$

Add to both sides:

$(14 p - 12) + 12 = 8 + 12$

Simplify the arithmetic:

$14 p = 8 + 12$

Simplify the arithmetic:

$14 p = 20$

Divide both sides by :

$\frac{(14 p)}{14} = \frac{20}{14}$

Simplify the fraction:

$p = \frac{20}{14}$

Find the greatest common factor of the numerator and denominator:

$p = \frac{(10 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$p = \frac{10}{7}$

15 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$16 p + 4 \cdot - 3 = - (2 p + 8)$

Simplify the arithmetic:

$16 p - 12 = - (2 p + 8)$

Expand the parentheses:

$16 p - 12 = - 2 p - 8$

Add to both sides:

$(16 p - 12) + 2 p = (- 2 p - 8) + 2 p$

Group like terms:

$(16 p + 2 p) - 12 = (- 2 p - 8) + 2 p$

Simplify the arithmetic:

$18 p - 12 = (- 2 p - 8) + 2 p$

Group like terms:

$18 p - 12 = (- 2 p + 2 p) - 8$

Simplify the arithmetic:

$18 p - 12 = - 8$

Add to both sides:

$(18 p - 12) + 12 = - 8 + 12$

Simplify the arithmetic:

$18 p = - 8 + 12$

Simplify the arithmetic:

$18 p = 4$

Divide both sides by :

$\frac{(18 p)}{18} = \frac{4}{18}$

Simplify the fraction:

$p = \frac{4}{18}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

$p = \frac{10}{7}, \frac{2}{9}$
(2 solution(s))

4. Graph

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