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Solution - Absolute value equations

Exact form:

p = - 4, - \frac{10}{9}

p=-4 , -\frac{10}{9}

Mixed number form:

p = - 4, - 1 \frac{1}{9}

p=-4 , -1\frac{1}{9}

Decimal form:

p = - 4, - 1.111

p=-4 , -1.111

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
$| 5 p + 7 | = | 4 p + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 p + 7 \| = \| 4 p + 3 \|$
$x = + y$	$(5 p + 7) = (4 p + 3)$
$x = - y$	$(5 p + 7) = - (4 p + 3)$
$+ x = y$	$(5 p + 7) = (4 p + 3)$
$- x = y$	$- (5 p + 7) = (4 p + 3)$

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:

$\| x \| = \| y \|$	$\| 5 p + 7 \| = \| 4 p + 3 \|$
$x = + y$ , $+ x = y$	$(5 p + 7) = (4 p + 3)$
$x = - y$ , $- x = y$	$(5 p + 7) = - (4 p + 3)$

2. Solve the two equations for p

7 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$p + 7 = 3$

Subtract from both sides:

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

Simplify the arithmetic:

$p = 3 - 7$

Simplify the arithmetic:

$p = - 4$

10 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$9 p + 7 = (- 4 p - 3) + 4 p$

Group like terms:

$9 p + 7 = (- 4 p + 4 p) - 3$

Simplify the arithmetic:

$9 p + 7 = - 3$

Subtract from both sides:

$(9 p + 7) - 7 = - 3 - 7$

Simplify the arithmetic:

$9 p = - 3 - 7$

Simplify the arithmetic:

$9 p = - 10$

Divide both sides by :

$\frac{(9 p)}{9} = \frac{- 10}{9}$

Simplify the fraction:

$p = \frac{- 10}{9}$

3. List the solutions

$p = - 4, - \frac{10}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 p + 7 |$
$y = | 4 p + 3 |$
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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