Solve for n 2|3n+3|<12

$2 | 3 n + 3 | < 12$

Divide each term in $2 | 3 n + 3 | < 12$ by $2$ and simplify.

Divide each term in $2 | 3 n + 3 | < 12$ by $2$ .

$\frac{2 | 3 n + 3 |}{2} < \frac{12}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 | 3 n + 3 |}{2} < \frac{12}{2}$

Divide $| 3 n + 3 |$ by $1$ .

$| 3 n + 3 | < \frac{12}{2}$

Simplify the right side.

Divide $12$ by $2$ .

$| 3 n + 3 | < 6$

Write $| 3 n + 3 | < 6$ as a piecewise.

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

$3 n + 3 \geq 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$3 n \geq - 3$

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

Divide each term in $3 n \geq - 3$ by $3$ .

$\frac{3 n}{3} \geq \frac{- 3}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 n}{3} \geq \frac{- 3}{3}$

Divide $n$ by $1$ .

$n \geq \frac{- 3}{3}$

Simplify the right side.

Divide $- 3$ by $3$ .

$n \geq - 1$

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

$3 n + 3 < 6$

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

$3 n + 3 < 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$3 n < - 3$

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

Divide each term in $3 n < - 3$ by $3$ .

$\frac{3 n}{3} < \frac{- 3}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 n}{3} < \frac{- 3}{3}$

Divide $n$ by $1$ .

$n < \frac{- 3}{3}$

Simplify the right side.

Divide $- 3$ by $3$ .

$n < - 1$

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

$- (3 n + 3) < 6$

Write as a piecewise.

${\begin{cases} 3 n + 3 < 6 & n \geq - 1 \\ - (3 n + 3) < 6 & n < - 1 \end{cases}$

Simplify $- (3 n + 3) < 6$ .

Apply the distributive property.

${\begin{cases} 3 n + 3 < 6 & n \geq - 1 \\ - (3 n) - 1 \cdot 3 < 6 & n < - 1 \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 3 n + 3 < 6 & n \geq - 1 \\ - 3 n - 1 \cdot 3 < 6 & n < - 1 \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} 3 n + 3 < 6 & n \geq - 1 \\ - 3 n - 3 < 6 & n < - 1 \end{cases}$

Solve $3 n + 3 < 6$ when $n \geq - 1$ .

Solve $3 n + 3 < 6$ for $n$ .

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

Subtract $3$ from both sides of the inequality.

$3 n < 6 - 3$

Subtract $3$ from $6$ .

$3 n < 3$

Divide each term in $3 n < 3$ by $3$ and simplify.

Divide each term in $3 n < 3$ by $3$ .

$\frac{3 n}{3} < \frac{3}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 n}{3} < \frac{3}{3}$

Divide $n$ by $1$ .

$n < \frac{3}{3}$

Simplify the right side.

Divide $3$ by $3$ .

$n < 1$

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

$- 1 \leq n < 1$

Solve $- 3 n - 3 < 6$ when $n < - 1$ .

Solve $- 3 n - 3 < 6$ for $n$ .

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

Add $3$ to both sides of the inequality.

$- 3 n < 6 + 3$

Add $6$ and $3$ .

$- 3 n < 9$

Divide each term in $- 3 n < 9$ by $- 3$ and simplify.

Divide each term in $- 3 n < 9$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 n}{- 3} > \frac{9}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 n}{- 3} > \frac{9}{- 3}$

Divide $n$ by $1$ .

$n > \frac{9}{- 3}$

Simplify the right side.

Divide $9$ by $- 3$ .

$n > - 3$

Find the intersection of $n > - 3$ and $n < - 1$ .

$- 3 < n < - 1$

Find the union of the solutions.

$- 3 < n < 1$

The result can be shown in multiple forms.

Inequality Form:

$- 3 < n < 1$

Interval Notation:

$(- 3, 1)$

Solve for n 2|3n+3|<12

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