Algebra Examples

$| - 3 p - 7 | > 5$

Step 1

Write $| - 3 p - 7 | > 5$ as a piecewise.

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Step 1.1

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

$- 3 p - 7 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Add $7$ to both sides of the inequality.

$- 3 p \geq 7$

Step 1.2.2

Divide each term in $- 3 p \geq 7$ by $- 3$ and simplify.

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Step 1.2.2.1

Divide each term in $- 3 p \geq 7$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 p}{- 3} \leq \frac{7}{- 3}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 p}{- 3} \leq \frac{7}{- 3}$

Step 1.2.2.2.1.2

Divide $p$ by $1$ .

$p \leq \frac{7}{- 3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Move the negative in front of the fraction.

$p \leq - \frac{7}{3}$

Step 1.3

In the piece where $- 3 p - 7$ is non-negative, remove the absolute value.

$- 3 p - 7 > 5$

Step 1.4

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

$- 3 p - 7 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Add $7$ to both sides of the inequality.

$- 3 p < 7$

Step 1.5.2

Divide each term in $- 3 p < 7$ by $- 3$ and simplify.

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Step 1.5.2.1

Divide each term in $- 3 p < 7$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 p}{- 3} > \frac{7}{- 3}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $- 3$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{- 3 p}{- 3} > \frac{7}{- 3}$

Step 1.5.2.2.1.2

Divide $p$ by $1$ .

$p > \frac{7}{- 3}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Move the negative in front of the fraction.

$p > - \frac{7}{3}$

Step 1.6

In the piece where $- 3 p - 7$ is negative, remove the absolute value and multiply by $- 1$ .

$- (- 3 p - 7) > 5$

Step 1.7

Write as a piecewise.

${\begin{cases} - 3 p - 7 > 5 & p \leq - \frac{7}{3} \\ - (- 3 p - 7) > 5 & p > - \frac{7}{3} \end{cases}$

Step 1.8

Simplify $- (- 3 p - 7) > 5$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} - 3 p - 7 > 5 & p \leq - \frac{7}{3} \\ - (- 3 p) - - 7 > 5 & p > - \frac{7}{3} \end{cases}$

Step 1.8.2

Multiply $- 3$ by $- 1$ .

${\begin{cases} - 3 p - 7 > 5 & p \leq - \frac{7}{3} \\ 3 p - - 7 > 5 & p > - \frac{7}{3} \end{cases}$

Step 1.8.3

Multiply $- 1$ by $- 7$ .

${\begin{cases} - 3 p - 7 > 5 & p \leq - \frac{7}{3} \\ 3 p + 7 > 5 & p > - \frac{7}{3} \end{cases}$

Step 2

Solve $- 3 p - 7 > 5$ for $p$ .

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Step 2.1

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

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Step 2.1.1

Add $7$ to both sides of the inequality.

$- 3 p > 5 + 7$

Step 2.1.2

Add $5$ and $7$ .

$- 3 p > 12$

Step 2.2

Divide each term in $- 3 p > 12$ by $- 3$ and simplify.

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Step 2.2.1

Divide each term in $- 3 p > 12$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 p}{- 3} < \frac{12}{- 3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 3 p}{- 3} < \frac{12}{- 3}$

Step 2.2.2.1.2

Divide $p$ by $1$ .

$p < \frac{12}{- 3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $12$ by $- 3$ .

$p < - 4$

Step 3

Solve $3 p + 7 > 5$ for $p$ .

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Step 3.1

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

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Step 3.1.1

Subtract $7$ from both sides of the inequality.

$3 p > 5 - 7$

Step 3.1.2

Subtract $7$ from $5$ .

$3 p > - 2$

Step 3.2

Divide each term in $3 p > - 2$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 p > - 2$ by $3$ .

$\frac{3 p}{3} > \frac{- 2}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 p}{3} > \frac{- 2}{3}$

Step 3.2.2.1.2

Divide $p$ by $1$ .

$p > \frac{- 2}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$p > - \frac{2}{3}$

Step 4

Find the union of the solutions.

$p < - 4$ or $p > - \frac{2}{3}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$p < - 4 or p > - \frac{2}{3}$

Interval Notation:

$(- \infty, - 4) \cup (- \frac{2}{3}, \infty)$

Step 6