Solve the Absolute Value Inequality for p |5-3p|+9=13p+8

$| 5 - 3 p | + 9 = 13 p + 8$

Step 1

Move all terms not containing $| 5 - 3 p |$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$| 5 - 3 p | = 13 p + 8 - 9$

Subtract $9$ from $8$ .

$| 5 - 3 p | = 13 p - 1$

Step 2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$5 - 3 p = \pm (13 p - 1)$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$5 - 3 p = 13 p - 1$

Move all terms containing $p$ to the left side of the equation.

Subtract $13 p$ from both sides of the equation.

$5 - 3 p - 13 p = - 1$

Subtract $13 p$ from $- 3 p$ .

$5 - 16 p = - 1$

Move all terms not containing $p$ to the right side of the equation.

Subtract $5$ from both sides of the equation.

$- 16 p = - 1 - 5$

Subtract $5$ from $- 1$ .

$- 16 p = - 6$

Divide each term in $- 16 p = - 6$ by $- 16$ and simplify.

Divide each term in $- 16 p = - 6$ by $- 16$ .

$\frac{- 16 p}{- 16} = \frac{- 6}{- 16}$

Simplify the left side.

Cancel the common factor of $- 16$ .

Cancel the common factor.

$\frac{- 16 p}{- 16} = \frac{- 6}{- 16}$

Divide $p$ by $1$ .

$p = \frac{- 6}{- 16}$

Simplify the right side.

Cancel the common factor of $- 6$ and $- 16$ .

Factor $- 2$ out of $- 6$ .

$p = \frac{- 2 (3)}{- 16}$

Cancel the common factors.

Factor $- 2$ out of $- 16$ .

$p = \frac{- 2 \cdot 3}{- 2 \cdot 8}$

Cancel the common factor.

$p = \frac{- 2 \cdot 3}{- 2 \cdot 8}$

Rewrite the expression.

$p = \frac{3}{8}$

Next, use the negative value of the $\pm$ to find the second solution.

$5 - 3 p = - (13 p - 1)$

Simplify $- (13 p - 1)$ .

Rewrite.

$5 - 3 p = 0 + 0 - (13 p - 1)$

Simplify by adding zeros.

$5 - 3 p = - (13 p - 1)$

Apply the distributive property.

$5 - 3 p = - (13 p) - - 1$

Multiply.

Multiply $13$ by $- 1$ .

$5 - 3 p = - 13 p - - 1$

Multiply $- 1$ by $- 1$ .

$5 - 3 p = - 13 p + 1$

Move all terms containing $p$ to the left side of the equation.

Add $13 p$ to both sides of the equation.

$5 - 3 p + 13 p = 1$

Add $- 3 p$ and $13 p$ .

$5 + 10 p = 1$

Move all terms not containing $p$ to the right side of the equation.

Subtract $5$ from both sides of the equation.

$10 p = 1 - 5$

Subtract $5$ from $1$ .

$10 p = - 4$

Divide each term in $10 p = - 4$ by $10$ and simplify.

Divide each term in $10 p = - 4$ by $10$ .

$\frac{10 p}{10} = \frac{- 4}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 p}{10} = \frac{- 4}{10}$

Divide $p$ by $1$ .

$p = \frac{- 4}{10}$

Simplify the right side.

Cancel the common factor of $- 4$ and $10$ .

Factor $2$ out of $- 4$ .

$p = \frac{2 (- 2)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$p = \frac{2 \cdot - 2}{2 \cdot 5}$

Cancel the common factor.

$p = \frac{2 \cdot - 2}{2 \cdot 5}$

Rewrite the expression.

$p = \frac{- 2}{5}$

Move the negative in front of the fraction.

$p = - \frac{2}{5}$

The complete solution is the result of both the positive and negative portions of the solution.

$p = \frac{3}{8}, - \frac{2}{5}$

Step 4

Exclude the solutions that do not make $| 5 - 3 p | + 9 = 13 p + 8$ true.

$p = \frac{3}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$p = \frac{3}{8}$

Decimal Form:

$p = 0.375$

Solve the Absolute Value Inequality for p |5-3p|+9=13p+8

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