Finite Math Examples

$| p | > 7$

Step 1

Write $| p | > 7$ 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.

$p \geq 0$

Step 1.2

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

$p > 7$

Step 1.3

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

$p < 0$

Step 1.4

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

$- p > 7$

Step 1.5

Write as a piecewise.

${\begin{cases} p > 7 & p \geq 0 \\ - p > 7 & p < 0 \end{cases}$

Step 2

Find the intersection of $p > 7$ and $p \geq 0$ .

$p > 7$

Step 3

Divide each term in $- p > 7$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- p > 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- p}{- 1} < \frac{7}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{p}{1} < \frac{7}{- 1}$

Step 3.2.2

Divide $p$ by $1$ .

$p < \frac{7}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $7$ by $- 1$ .

$p < - 7$

Step 4

Find the union of the solutions.

$p < - 7$ or $p > 7$

Step 5

Convert the inequality to interval notation.

$(- \infty, - 7) \cup (7, \infty)$

Step 6