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Solution - Absolute value equations

Exact form:

p = \frac{1}{36}, - \frac{17}{72}

p=\frac{1}{36} , -\frac{17}{72}

Decimal form:

p = 0.028, - 0.236

p=0.028 , -0.236

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
$| 3 p + \frac{4}{9} | = | p + \frac{1}{2} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 p + \frac{4}{9} \| = \| p + \frac{1}{2} \|$
$x = + y$	$(3 p + \frac{4}{9}) = (p + \frac{1}{2})$
$x = - y$	$(3 p + \frac{4}{9}) = - (p + \frac{1}{2})$
$+ x = y$	$(3 p + \frac{4}{9}) = (p + \frac{1}{2})$
$- x = y$	$- (3 p + \frac{4}{9}) = (p + \frac{1}{2})$

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:

$\| x \| = \| y \|$	$\| 3 p + \frac{4}{9} \| = \| p + \frac{1}{2} \|$
$x = + y$ , $+ x = y$	$(3 p + \frac{4}{9}) = (p + \frac{1}{2})$
$x = - y$ , $- x = y$	$(3 p + \frac{4}{9}) = - (p + \frac{1}{2})$

2. Solve the two equations for p

18 additional steps

$(3 p + \frac{4}{9}) = (p + \frac{1}{2})$

Subtract from both sides:

$(3 p + \frac{4}{9}) - p = (p + \frac{1}{2}) - p$

Group like terms:

$(3 p - p) + \frac{4}{9} = (p + \frac{1}{2}) - p$

Simplify the arithmetic:

$2 p + \frac{4}{9} = (p + \frac{1}{2}) - p$

Group like terms:

$2 p + \frac{4}{9} = (p - p) + \frac{1}{2}$

Simplify the arithmetic:

$2 p + \frac{4}{9} = \frac{1}{2}$

Subtract from both sides:

$(2 p + \frac{4}{9}) - \frac{4}{9} = (\frac{1}{2}) - \frac{4}{9}$

Combine the fractions:

$2 p + \frac{(4 - 4)}{9} = (\frac{1}{2}) - \frac{4}{9}$

Combine the numerators:

$2 p + \frac{0}{9} = (\frac{1}{2}) - \frac{4}{9}$

Reduce the zero numerator:

$2 p + 0 = (\frac{1}{2}) - \frac{4}{9}$

Simplify the arithmetic:

$2 p = (\frac{1}{2}) - \frac{4}{9}$

Find the lowest common denominator:

$2 p = \frac{(1 \cdot 9)}{(2 \cdot 9)} + \frac{(- 4 \cdot 2)}{(9 \cdot 2)}$

Multiply the denominators:

$2 p = \frac{(1 \cdot 9)}{18} + \frac{(- 4 \cdot 2)}{18}$

Multiply the numerators:

$2 p = \frac{9}{18} + \frac{- 8}{18}$

Combine the fractions:

$2 p = \frac{(9 - 8)}{18}$

Combine the numerators:

$2 p = \frac{1}{18}$

Divide both sides by :

$\frac{(2 p)}{2} = \frac{(\frac{1}{18})}{2}$

Simplify the fraction:

$p = \frac{(\frac{1}{18})}{2}$

Simplify the arithmetic:

$p = \frac{1}{(18 \cdot 2)}$

$p = \frac{1}{36}$

19 additional steps

$(3 p + \frac{4}{9}) = - (p + \frac{1}{2})$

Expand the parentheses:

$(3 p + \frac{4}{9}) = - p + \frac{- 1}{2}$

Add to both sides:

$(3 p + \frac{4}{9}) + p = (- p + \frac{- 1}{2}) + p$

Group like terms:

$(3 p + p) + \frac{4}{9} = (- p + \frac{- 1}{2}) + p$

Simplify the arithmetic:

$4 p + \frac{4}{9} = (- p + \frac{- 1}{2}) + p$

Group like terms:

$4 p + \frac{4}{9} = (- p + p) + \frac{- 1}{2}$

Simplify the arithmetic:

$4 p + \frac{4}{9} = \frac{- 1}{2}$

Subtract from both sides:

$(4 p + \frac{4}{9}) - \frac{4}{9} = (\frac{- 1}{2}) - \frac{4}{9}$

Combine the fractions:

$4 p + \frac{(4 - 4)}{9} = (\frac{- 1}{2}) - \frac{4}{9}$

Combine the numerators:

$4 p + \frac{0}{9} = (\frac{- 1}{2}) - \frac{4}{9}$

Reduce the zero numerator:

$4 p + 0 = (\frac{- 1}{2}) - \frac{4}{9}$

Simplify the arithmetic:

$4 p = (\frac{- 1}{2}) - \frac{4}{9}$

Find the lowest common denominator:

$4 p = \frac{(- 1 \cdot 9)}{(2 \cdot 9)} + \frac{(- 4 \cdot 2)}{(9 \cdot 2)}$

Multiply the denominators:

$4 p = \frac{(- 1 \cdot 9)}{18} + \frac{(- 4 \cdot 2)}{18}$

Multiply the numerators:

$4 p = \frac{- 9}{18} + \frac{- 8}{18}$

Combine the fractions:

$4 p = \frac{(- 9 - 8)}{18}$

Combine the numerators:

$4 p = \frac{- 17}{18}$

Divide both sides by :

$\frac{(4 p)}{4} = \frac{(\frac{- 17}{18})}{4}$

Simplify the fraction:

$p = \frac{(\frac{- 17}{18})}{4}$

Simplify the arithmetic:

$p = \frac{- 17}{(18 \cdot 4)}$

$p = \frac{- 17}{72}$

3. List the solutions

$p = \frac{1}{36}, - \frac{17}{72}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 p + \frac{4}{9} |$
$y = | p + \frac{1}{2} |$
The equation is true where the two lines cross.

How did we do?

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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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