Solve for p |p+3|>=10

$| p + 3 | \geq 10$

Step 1

Write $| p + 3 | \geq 10$ as a piecewise.

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

$p + 3 \geq 0$

Subtract $3$ from both sides of the inequality.

$p \geq - 3$

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

$p + 3 \geq 10$

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

$p + 3 < 0$

Subtract $3$ from both sides of the inequality.

$p < - 3$

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

$- (p + 3) \geq 10$

Write as a piecewise.

${\begin{cases} p + 3 \geq 10 & p \geq - 3 \\ - (p + 3) \geq 10 & p < - 3 \end{cases}$

Simplify $- (p + 3) \geq 10$ .

Apply the distributive property.

${\begin{cases} p + 3 \geq 10 & p \geq - 3 \\ - p - 1 \cdot 3 \geq 10 & p < - 3 \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} p + 3 \geq 10 & p \geq - 3 \\ - p - 3 \geq 10 & p < - 3 \end{cases}$

Step 2

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

Subtract $3$ from both sides of the inequality.

$p \geq 10 - 3$

Subtract $3$ from $10$ .

$p \geq 7$

Step 3

Solve $- p - 3 \geq 10$ for $p$ .

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

Add $3$ to both sides of the inequality.

$- p \geq 10 + 3$

Add $10$ and $3$ .

$- p \geq 13$

Divide each term in $- p \geq 13$ by $- 1$ and simplify.

Divide each term in $- p \geq 13$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- p}{- 1} \leq \frac{13}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{p}{1} \leq \frac{13}{- 1}$

Divide $p$ by $1$ .

$p \leq \frac{13}{- 1}$

Simplify the right side.

Divide $13$ by $- 1$ .

$p \leq - 13$

Step 4

Find the union of the solutions.

$p \leq - 13$ or $p \geq 7$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$p \leq - 13 or p \geq 7$

Interval Notation:

$(- \infty, - 13] \cup [7, \infty)$

Step 6

Solve for p |p+3|>=10

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