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Solution - Absolute value equations

Exact form:

p = \frac{3}{4}

p=\frac{3}{4}

Decimal form:

p = 0.75

p=0.75

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
$| - 8 p - 3 | = | - 8 p + 15 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 8 p - 3 \| = \| - 8 p + 15 \|$
$x = + y$	$(- 8 p - 3) = (- 8 p + 15)$
$x = - y$	$(- 8 p - 3) = - (- 8 p + 15)$
$+ x = y$	$(- 8 p - 3) = (- 8 p + 15)$
$- x = y$	$- (- 8 p - 3) = (- 8 p + 15)$

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:

$\| x \| = \| y \|$	$\| - 8 p - 3 \| = \| - 8 p + 15 \|$
$x = + y$ , $+ x = y$	$(- 8 p - 3) = (- 8 p + 15)$
$x = - y$ , $- x = y$	$(- 8 p - 3) = - (- 8 p + 15)$

2. Solve the two equations for p

5 additional steps

$(- 8 p - 3) = (- 8 p + 15)$

Add to both sides:

$(- 8 p - 3) + 8 p = (- 8 p + 15) + 8 p$

Group like terms:

$(- 8 p + 8 p) - 3 = (- 8 p + 15) + 8 p$

Simplify the arithmetic:

$- 3 = (- 8 p + 15) + 8 p$

Group like terms:

$- 3 = (- 8 p + 8 p) + 15$

Simplify the arithmetic:

$- 3 = 15$

The statement is false:

$- 3 = 15$

The equation is false so it has no solution.

14 additional steps

$(- 8 p - 3) = - (- 8 p + 15)$

Expand the parentheses:

$(- 8 p - 3) = 8 p - 15$

Subtract from both sides:

$(- 8 p - 3) - 8 p = (8 p - 15) - 8 p$

Group like terms:

$(- 8 p - 8 p) - 3 = (8 p - 15) - 8 p$

Simplify the arithmetic:

$- 16 p - 3 = (8 p - 15) - 8 p$

Group like terms:

$- 16 p - 3 = (8 p - 8 p) - 15$

Simplify the arithmetic:

$- 16 p - 3 = - 15$

Add to both sides:

$(- 16 p - 3) + 3 = - 15 + 3$

Simplify the arithmetic:

$- 16 p = - 15 + 3$

Simplify the arithmetic:

$- 16 p = - 12$

Divide both sides by :

$\frac{(- 16 p)}{- 16} = \frac{- 12}{- 16}$

Cancel out the negatives:

$\frac{16 p}{16} = \frac{- 12}{- 16}$

Simplify the fraction:

$p = \frac{- 12}{- 16}$

Cancel out the negatives:

$p = \frac{12}{16}$

Find the greatest common factor of the numerator and denominator:

$p = \frac{(3 \cdot 4)}{(4 \cdot 4)}$

Factor out and cancel the greatest common factor:

$p = \frac{3}{4}$

3. Graph

Each line represents the function of one side of the equation:
$y = | - 8 p - 3 |$
$y = | - 8 p + 15 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved