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Solution - Absolute value equations

Exact form:

p = - 6, - \frac{1}{2}

p=-6 , -\frac{1}{2}

Decimal form:

p = - 6, - 0.5

p=-6 , -0.5

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
$| p - 5 | = | 3 p + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| p - 5 \| = \| 3 p + 7 \|$
$x = + y$	$(p - 5) = (3 p + 7)$
$x = - y$	$(p - 5) = - (3 p + 7)$
$+ x = y$	$(p - 5) = (3 p + 7)$
$- x = y$	$- (p - 5) = (3 p + 7)$

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:

$\| x \| = \| y \|$	$\| p - 5 \| = \| 3 p + 7 \|$
$x = + y$ , $+ x = y$	$(p - 5) = (3 p + 7)$
$x = - y$ , $- x = y$	$(p - 5) = - (3 p + 7)$

2. Solve the two equations for p

13 additional steps

$(p - 5) = (3 p + 7)$

Subtract from both sides:

$(p - 5) - 3 p = (3 p + 7) - 3 p$

Group like terms:

$(p - 3 p) - 5 = (3 p + 7) - 3 p$

Simplify the arithmetic:

$- 2 p - 5 = (3 p + 7) - 3 p$

Group like terms:

$- 2 p - 5 = (3 p - 3 p) + 7$

Simplify the arithmetic:

$- 2 p - 5 = 7$

Add to both sides:

$(- 2 p - 5) + 5 = 7 + 5$

Simplify the arithmetic:

$- 2 p = 7 + 5$

Simplify the arithmetic:

$- 2 p = 12$

Divide both sides by :

$\frac{(- 2 p)}{- 2} = \frac{12}{- 2}$

Cancel out the negatives:

$\frac{2 p}{2} = \frac{12}{- 2}$

Simplify the fraction:

$p = \frac{12}{- 2}$

Move the negative sign from the denominator to the numerator:

$p = \frac{- 12}{2}$

Find the greatest common factor of the numerator and denominator:

$p = \frac{(- 6 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$p = - 6$

12 additional steps

$(p - 5) = - (3 p + 7)$

Expand the parentheses:

$(p - 5) = - 3 p - 7$

Add to both sides:

$(p - 5) + 3 p = (- 3 p - 7) + 3 p$

Group like terms:

$(p + 3 p) - 5 = (- 3 p - 7) + 3 p$

Simplify the arithmetic:

$4 p - 5 = (- 3 p - 7) + 3 p$

Group like terms:

$4 p - 5 = (- 3 p + 3 p) - 7$

Simplify the arithmetic:

$4 p - 5 = - 7$

Add to both sides:

$(4 p - 5) + 5 = - 7 + 5$

Simplify the arithmetic:

$4 p = - 7 + 5$

Simplify the arithmetic:

$4 p = - 2$

Divide both sides by :

$\frac{(4 p)}{4} = \frac{- 2}{4}$

Simplify the fraction:

$p = \frac{- 2}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$p = \frac{- 1}{2}$

3. List the solutions

$p = - 6, - \frac{1}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | p - 5 |$
$y = | 3 p + 7 |$
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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