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Solution - Absolute value equations

Exact form:

p = 51, - 3

p=51 , -3

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
$| 5 p + 42 | = | 6 p - 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 p + 42 \| = \| 6 p - 9 \|$
$x = + y$	$(5 p + 42) = (6 p - 9)$
$x = - y$	$(5 p + 42) = - (6 p - 9)$
$+ x = y$	$(5 p + 42) = (6 p - 9)$
$- x = y$	$- (5 p + 42) = (6 p - 9)$

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:

$\| x \| = \| y \|$	$\| 5 p + 42 \| = \| 6 p - 9 \|$
$x = + y$ , $+ x = y$	$(5 p + 42) = (6 p - 9)$
$x = - y$ , $- x = y$	$(5 p + 42) = - (6 p - 9)$

2. Solve the two equations for p

10 additional steps

$(5 p + 42) = (6 p - 9)$

Subtract from both sides:

$(5 p + 42) - 6 p = (6 p - 9) - 6 p$

Group like terms:

$(5 p - 6 p) + 42 = (6 p - 9) - 6 p$

Simplify the arithmetic:

$- p + 42 = (6 p - 9) - 6 p$

Group like terms:

$- p + 42 = (6 p - 6 p) - 9$

Simplify the arithmetic:

$- p + 42 = - 9$

Subtract from both sides:

$(- p + 42) - 42 = - 9 - 42$

Simplify the arithmetic:

$- p = - 9 - 42$

Simplify the arithmetic:

$- p = - 51$

Multiply both sides by :

$- p \cdot - 1 = - 51 \cdot - 1$

Remove the one(s):

$p = - 51 \cdot - 1$

Simplify the arithmetic:

$p = 51$

12 additional steps

$(5 p + 42) = - (6 p - 9)$

Expand the parentheses:

$(5 p + 42) = - 6 p + 9$

Add to both sides:

$(5 p + 42) + 6 p = (- 6 p + 9) + 6 p$

Group like terms:

$(5 p + 6 p) + 42 = (- 6 p + 9) + 6 p$

Simplify the arithmetic:

$11 p + 42 = (- 6 p + 9) + 6 p$

Group like terms:

$11 p + 42 = (- 6 p + 6 p) + 9$

Simplify the arithmetic:

$11 p + 42 = 9$

Subtract from both sides:

$(11 p + 42) - 42 = 9 - 42$

Simplify the arithmetic:

$11 p = 9 - 42$

Simplify the arithmetic:

$11 p = - 33$

Divide both sides by :

$\frac{(11 p)}{11} = \frac{- 33}{11}$

Simplify the fraction:

$p = \frac{- 33}{11}$

Find the greatest common factor of the numerator and denominator:

$p = \frac{(- 3 \cdot 11)}{(1 \cdot 11)}$

Factor out and cancel the greatest common factor:

$p = - 3$

3. List the solutions

$p = 51, - 3$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 p + 42 |$
$y = | 6 p - 9 |$
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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