Solve for n n/9<n/2-1/3

$\frac{n}{9} < \frac{n}{2} - \frac{1}{3}$

Step 1

Multiply both sides by $9$ .

$\frac{n}{9} \cdot 9 < (\frac{n}{2} - \frac{1}{3}) \cdot 9$

Step 2

Simplify.

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{n}{9} \cdot 9 < (\frac{n}{2} - \frac{1}{3}) \cdot 9$

Rewrite the expression.

$n < (\frac{n}{2} - \frac{1}{3}) \cdot 9$

Simplify the right side.

Simplify $(\frac{n}{2} - \frac{1}{3}) \cdot 9$ .

Simplify terms.

Apply the distributive property.

$n < \frac{n}{2} \cdot 9 - \frac{1}{3} \cdot 9$

Combine $\frac{n}{2}$ and $9$ .

$n < \frac{n \cdot 9}{2} - \frac{1}{3} \cdot 9$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$n < \frac{n \cdot 9}{2} + \frac{- 1}{3} \cdot 9$

Factor $3$ out of $9$ .

$n < \frac{n \cdot 9}{2} + \frac{- 1}{3} \cdot (3 (3))$

Cancel the common factor.

$n < \frac{n \cdot 9}{2} + \frac{- 1}{3} \cdot (3 \cdot 3)$

Rewrite the expression.

$n < \frac{n \cdot 9}{2} - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$n < \frac{n \cdot 9}{2} - 3$

Move $9$ to the left of $n$ .

$n < \frac{9 n}{2} - 3$

Step 3

Solve for $n$ .

Move all terms containing $n$ to the left side of the inequality.

Subtract $\frac{9 n}{2}$ from both sides of the inequality.

$n - \frac{9 n}{2} < - 3$

To write $n$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$n \cdot \frac{2}{2} - \frac{9 n}{2} < - 3$

Combine $n$ and $\frac{2}{2}$ .

$\frac{n \cdot 2}{2} - \frac{9 n}{2} < - 3$

Combine the numerators over the common denominator.

$\frac{n \cdot 2 - 9 n}{2} < - 3$

Simplify the numerator.

Move $2$ to the left of $n$ .

$\frac{2 \cdot n - 9 n}{2} < - 3$

Subtract $9 n$ from $2 n$ .

$\frac{- 7 n}{2} < - 3$

Move the negative in front of the fraction.

$- \frac{7 n}{2} < - 3$

Divide each term in $- \frac{7 n}{2} < - 3$ by $- 1$ and simplify.

Divide each term in $- \frac{7 n}{2} < - 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{7 n}{2}}{- 1} > \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{7 n}{2}}{1} > \frac{- 3}{- 1}$

Divide $\frac{7 n}{2}$ by $1$ .

$\frac{7 n}{2} > \frac{- 3}{- 1}$

Simplify the right side.

Divide $- 3$ by $- 1$ .

$\frac{7 n}{2} > 3$

Multiply both sides by $2$ .

$\frac{7 n}{2} \cdot 2 > 3 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{7 n}{2} \cdot 2 > 3 \cdot 2$

Rewrite the expression.

$7 n > 3 \cdot 2$

Simplify the right side.

Multiply $3$ by $2$ .

$7 n > 6$

Divide each term in $7 n > 6$ by $7$ and simplify.

Divide each term in $7 n > 6$ by $7$ .

$\frac{7 n}{7} > \frac{6}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 n}{7} > \frac{6}{7}$

Divide $n$ by $1$ .

$n > \frac{6}{7}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$n > \frac{6}{7}$

Interval Notation:

$(\frac{6}{7}, \infty)$

Solve for n n/9<n/2-1/3

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