Solve the Inequality for n 3|n-1|>9

$3 | n - 1 | > 9$

Step 1

Write $3 | n - 1 | > 9$ as a piecewise.

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

$n - 1 \geq 0$

Add $1$ to both sides of the inequality.

$n \geq 1$

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

$3 (n - 1) > 9$

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

$n - 1 < 0$

Add $1$ to both sides of the inequality.

$n < 1$

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

$3 (- (n - 1)) > 9$

Write as a piecewise.

${\begin{cases} 3 (n - 1) > 9 & n \geq 1 \\ 3 (- (n - 1)) > 9 & n < 1 \end{cases}$

Simplify $3 (n - 1) > 9$ .

Apply the distributive property.

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

Multiply $3$ by $- 1$ .

${\begin{cases} 3 n - 3 > 9 & n \geq 1 \\ 3 (- (n - 1)) > 9 & n < 1 \end{cases}$

Simplify $3 (- (n - 1)) > 9$ .

Apply the distributive property.

${\begin{cases} 3 n - 3 > 9 & n \geq 1 \\ 3 (- n - - 1) > 9 & n < 1 \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} 3 n - 3 > 9 & n \geq 1 \\ 3 (- n + 1) > 9 & n < 1 \end{cases}$

Apply the distributive property.

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

Multiply $- 1$ by $3$ .

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

Multiply $3$ by $1$ .

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

Step 2

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

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

Add $3$ to both sides of the inequality.

$3 n > 9 + 3$

Add $9$ and $3$ .

$3 n > 12$

Divide each term in $3 n > 12$ by $3$ and simplify.

Divide each term in $3 n > 12$ by $3$ .

$\frac{3 n}{3} > \frac{12}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 n}{3} > \frac{12}{3}$

Divide $n$ by $1$ .

$n > \frac{12}{3}$

Simplify the right side.

Divide $12$ by $3$ .

$n > 4$

Step 3

Solve $- 3 n + 3 > 9$ for $n$ .

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

Subtract $3$ from both sides of the inequality.

$- 3 n > 9 - 3$

Subtract $3$ from $9$ .

$- 3 n > 6$

Divide each term in $- 3 n > 6$ by $- 3$ and simplify.

Divide each term in $- 3 n > 6$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 n}{- 3} < \frac{6}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 n}{- 3} < \frac{6}{- 3}$

Divide $n$ by $1$ .

$n < \frac{6}{- 3}$

Simplify the right side.

Divide $6$ by $- 3$ .

$n < - 2$

Step 4

Find the union of the solutions.

$n < - 2$ or $n > 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$n < - 2 or n > 4$

Interval Notation:

$(- \infty, - 2) \cup (4, \infty)$

Step 6

Solve the Inequality for n 3|n-1|>9

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