Solve for p |10p-4|<34

$| 10 p - 4 | < 34$

Step 1

Write $| 10 p - 4 | < 34$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$10 p - 4 \geq 0$

Solve the inequality.

Add $4$ to both sides of the inequality.

$10 p \geq 4$

Divide each term in $10 p \geq 4$ by $10$ and simplify.

Divide each term in $10 p \geq 4$ by $10$ .

$\frac{10 p}{10} \geq \frac{4}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 p}{10} \geq \frac{4}{10}$

Divide $p$ by $1$ .

$p \geq \frac{4}{10}$

Simplify the right side.

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

Factor $2$ out of $4$ .

$p \geq \frac{2 (2)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$p \geq \frac{2 \cdot 2}{2 \cdot 5}$

Cancel the common factor.

$p \geq \frac{2 \cdot 2}{2 \cdot 5}$

Rewrite the expression.

$p \geq \frac{2}{5}$

In the piece where $10 p - 4$ is non-negative, remove the absolute value.

$10 p - 4 < 34$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$10 p - 4 < 0$

Solve the inequality.

Add $4$ to both sides of the inequality.

$10 p < 4$

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.

$p < \frac{2}{5}$

In the piece where $10 p - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (10 p - 4) < 34$

Write as a piecewise.

${\begin{cases} 10 p - 4 < 34 & p \geq \frac{2}{5} \\ - (10 p - 4) < 34 & p < \frac{2}{5} \end{cases}$

Simplify $- (10 p - 4) < 34$ .

Apply the distributive property.

${\begin{cases} 10 p - 4 < 34 & p \geq \frac{2}{5} \\ - (10 p) - - 4 < 34 & p < \frac{2}{5} \end{cases}$

Multiply $10$ by $- 1$ .

${\begin{cases} 10 p - 4 < 34 & p \geq \frac{2}{5} \\ - 10 p - - 4 < 34 & p < \frac{2}{5} \end{cases}$

Multiply $- 1$ by $- 4$ .

${\begin{cases} 10 p - 4 < 34 & p \geq \frac{2}{5} \\ - 10 p + 4 < 34 & p < \frac{2}{5} \end{cases}$

Step 2

Solve $10 p - 4 < 34$ when $p \geq \frac{2}{5}$ .

Solve $10 p - 4 < 34$ for $p$ .

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

Add $4$ to both sides of the inequality.

$10 p < 34 + 4$

Add $34$ and $4$ .

$10 p < 38$

Divide each term in $10 p < 38$ by $10$ and simplify.

Divide each term in $10 p < 38$ by $10$ .

$\frac{10 p}{10} < \frac{38}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 p}{10} < \frac{38}{10}$

Divide $p$ by $1$ .

$p < \frac{38}{10}$

Simplify the right side.

Cancel the common factor of $38$ and $10$ .

Factor $2$ out of $38$ .

$p < \frac{2 (19)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$p < \frac{2 \cdot 19}{2 \cdot 5}$

Cancel the common factor.

$p < \frac{2 \cdot 19}{2 \cdot 5}$

Rewrite the expression.

$p < \frac{19}{5}$

Find the intersection of $p < \frac{19}{5}$ and $p \geq \frac{2}{5}$ .

$\frac{2}{5} \leq p < \frac{19}{5}$

Step 3

Solve $- 10 p + 4 < 34$ when $p < \frac{2}{5}$ .

Solve $- 10 p + 4 < 34$ for $p$ .

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

Subtract $4$ from both sides of the inequality.

$- 10 p < 34 - 4$

Subtract $4$ from $34$ .

$- 10 p < 30$

Divide each term in $- 10 p < 30$ by $- 10$ and simplify.

Divide each term in $- 10 p < 30$ by $- 10$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 10 p}{- 10} > \frac{30}{- 10}$

Simplify the left side.

Cancel the common factor of $- 10$ .

Cancel the common factor.

$\frac{- 10 p}{- 10} > \frac{30}{- 10}$

Divide $p$ by $1$ .

$p > \frac{30}{- 10}$

Simplify the right side.

Divide $30$ by $- 10$ .

$p > - 3$

Find the intersection of $p > - 3$ and $p < \frac{2}{5}$ .

$- 3 < p < \frac{2}{5}$

Step 4

Find the union of the solutions.

$- 3 < p < \frac{19}{5}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 3 < p < \frac{19}{5}$

Interval Notation:

$(- 3, \frac{19}{5})$

Step 6

Solve for p |10p-4|<34

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