Solve the Inequality for d |d+1|<=6

$| d + 1 | \leq 6$

Step 1

Write $| d + 1 | \leq 6$ as a piecewise.

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

$d + 1 \geq 0$

Subtract $1$ from both sides of the inequality.

$d \geq - 1$

In the piece where $d + 1$ is non-negative, remove the absolute value.

$d + 1 \leq 6$

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

$d + 1 < 0$

Subtract $1$ from both sides of the inequality.

$d < - 1$

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

$- (d + 1) \leq 6$

Write as a piecewise.

${\begin{cases} d + 1 \leq 6 & d \geq - 1 \\ - (d + 1) \leq 6 & d < - 1 \end{cases}$

Simplify $- (d + 1) \leq 6$ .

Apply the distributive property.

${\begin{cases} d + 1 \leq 6 & d \geq - 1 \\ - d - 1 \cdot 1 \leq 6 & d < - 1 \end{cases}$

Multiply $- 1$ by $1$ .

${\begin{cases} d + 1 \leq 6 & d \geq - 1 \\ - d - 1 \leq 6 & d < - 1 \end{cases}$

Step 2

Solve $d + 1 \leq 6$ when $d \geq - 1$ .

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

Subtract $1$ from both sides of the inequality.

$d \leq 6 - 1$

Subtract $1$ from $6$ .

$d \leq 5$

Find the intersection of $d \leq 5$ and $d \geq - 1$ .

$- 1 \leq d \leq 5$

Step 3

Solve $- d - 1 \leq 6$ when $d < - 1$ .

Solve $- d - 1 \leq 6$ for $d$ .

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

Add $1$ to both sides of the inequality.

$- d \leq 6 + 1$

Add $6$ and $1$ .

$- d \leq 7$

Divide each term in $- d \leq 7$ by $- 1$ and simplify.

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

$\frac{- d}{- 1} \geq \frac{7}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{d}{1} \geq \frac{7}{- 1}$

Divide $d$ by $1$ .

$d \geq \frac{7}{- 1}$

Simplify the right side.

Divide $7$ by $- 1$ .

$d \geq - 7$

Find the intersection of $d \geq - 7$ and $d < - 1$ .

$- 7 \leq d < - 1$

Step 4

Find the union of the solutions.

$- 7 \leq d \leq 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 7 \leq d \leq 5$

Interval Notation:

$[- 7, 5]$

Step 6

Solve the Inequality for d |d+1|<=6

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