Solve the Inequality for d |d+9|>3

$| d + 9 | > 3$

Step 1

Write $| d + 9 | > 3$ as a piecewise.

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

$d + 9 \geq 0$

Subtract $9$ from both sides of the inequality.

$d \geq - 9$

In the piece where $d + 9$ is non-negative, remove the absolute value.

$d + 9 > 3$

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

$d + 9 < 0$

Subtract $9$ from both sides of the inequality.

$d < - 9$

In the piece where $d + 9$ is negative, remove the absolute value and multiply by $- 1$ .

$- (d + 9) > 3$

Write as a piecewise.

${\begin{cases} d + 9 > 3 & d \geq - 9 \\ - (d + 9) > 3 & d < - 9 \end{cases}$

Simplify $- (d + 9) > 3$ .

Apply the distributive property.

${\begin{cases} d + 9 > 3 & d \geq - 9 \\ - d - 1 \cdot 9 > 3 & d < - 9 \end{cases}$

Multiply $- 1$ by $9$ .

${\begin{cases} d + 9 > 3 & d \geq - 9 \\ - d - 9 > 3 & d < - 9 \end{cases}$

Step 2

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

Subtract $9$ from both sides of the inequality.

$d > 3 - 9$

Subtract $9$ from $3$ .

$d > - 6$

Step 3

Solve $- d - 9 > 3$ for $d$ .

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

Add $9$ to both sides of the inequality.

$- d > 3 + 9$

Add $3$ and $9$ .

$- d > 12$

Divide each term in $- d > 12$ by $- 1$ and simplify.

Divide each term in $- d > 12$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- d}{- 1} < \frac{12}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{d}{1} < \frac{12}{- 1}$

Divide $d$ by $1$ .

$d < \frac{12}{- 1}$

Simplify the right side.

Divide $12$ by $- 1$ .

$d < - 12$

Step 4

Find the union of the solutions.

$d < - 12$ or $d > - 6$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$d < - 12 or d > - 6$

Interval Notation:

$(- \infty, - 12) \cup (- 6, \infty)$

Step 6

Solve the Inequality for d |d+9|>3

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