Solve the Inequality for n (n-2)/5-n/3<=2

$\frac{n - 2}{5} - \frac{n}{3} \leq 2$

Step 1

Simplify $\frac{n - 2}{5} - \frac{n}{3}$ .

To write $\frac{n - 2}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{n - 2}{5} \cdot \frac{3}{3} - \frac{n}{3} \leq 2$

To write $- \frac{n}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{n - 2}{5} \cdot \frac{3}{3} - \frac{n}{3} \cdot \frac{5}{5} \leq 2$

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{n - 2}{5}$ by $\frac{3}{3}$ .

$\frac{(n - 2) \cdot 3}{5 \cdot 3} - \frac{n}{3} \cdot \frac{5}{5} \leq 2$

Multiply $5$ by $3$ .

$\frac{(n - 2) \cdot 3}{15} - \frac{n}{3} \cdot \frac{5}{5} \leq 2$

Multiply $\frac{n}{3}$ by $\frac{5}{5}$ .

$\frac{(n - 2) \cdot 3}{15} - \frac{n \cdot 5}{3 \cdot 5} \leq 2$

Multiply $3$ by $5$ .

$\frac{(n - 2) \cdot 3}{15} - \frac{n \cdot 5}{15} \leq 2$

Combine the numerators over the common denominator.

$\frac{(n - 2) \cdot 3 - n \cdot 5}{15} \leq 2$

Simplify the numerator.

Apply the distributive property.

$\frac{n \cdot 3 - 2 \cdot 3 - n \cdot 5}{15} \leq 2$

Move $3$ to the left of $n$ .

$\frac{3 \cdot n - 2 \cdot 3 - n \cdot 5}{15} \leq 2$

Multiply $- 2$ by $3$ .

$\frac{3 \cdot n - 6 - n \cdot 5}{15} \leq 2$

Multiply $5$ by $- 1$ .

$\frac{3 n - 6 - 5 n}{15} \leq 2$

Subtract $5 n$ from $3 n$ .

$\frac{- 2 n - 6}{15} \leq 2$

Factor $2$ out of $- 2 n - 6$ .

Factor $2$ out of $- 2 n$ .

$\frac{2 (- n) - 6}{15} \leq 2$

Factor $2$ out of $- 6$ .

$\frac{2 (- n) + 2 (- 3)}{15} \leq 2$

Factor $2$ out of $2 (- n) + 2 (- 3)$ .

$\frac{2 (- n - 3)}{15} \leq 2$

Simplify with factoring out.

Factor $- 1$ out of $- n$ .

$\frac{2 (- (n) - 3)}{15} \leq 2$

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{2 (- (n) - 1 (3))}{15} \leq 2$

Factor $- 1$ out of $- (n) - 1 (3)$ .

$\frac{2 (- (n + 3))}{15} \leq 2$

Simplify the expression.

Rewrite $- (n + 3)$ as $- 1 (n + 3)$ .

$\frac{2 (- 1 (n + 3))}{15} \leq 2$

Move the negative in front of the fraction.

$- \frac{2 (n + 3)}{15} \leq 2$

Step 2

Divide each term in $- \frac{2 (n + 3)}{15} \leq 2$ by $- 1$ and simplify.

Divide each term in $- \frac{2 (n + 3)}{15} \leq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{2 (n + 3)}{15}}{- 1} \geq \frac{2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{2 (n + 3)}{15}}{1} \geq \frac{2}{- 1}$

Divide $\frac{2 (n + 3)}{15}$ by $1$ .

$\frac{2 (n + 3)}{15} \geq \frac{2}{- 1}$

Simplify the right side.

Divide $2$ by $- 1$ .

$\frac{2 (n + 3)}{15} \geq - 2$

Step 3

Multiply both sides by $15$ .

$\frac{2 (n + 3)}{15} \cdot 15 \geq - 2 \cdot 15$

Step 4

Simplify.

Simplify the left side.

Simplify $\frac{2 (n + 3)}{15} \cdot 15$ .

Cancel the common factor of $15$ .

Cancel the common factor.

$\frac{2 (n + 3)}{15} \cdot 15 \geq - 2 \cdot 15$

Rewrite the expression.

$2 (n + 3) \geq - 2 \cdot 15$

Apply the distributive property.

$2 n + 2 \cdot 3 \geq - 2 \cdot 15$

Multiply $2$ by $3$ .

$2 n + 6 \geq - 2 \cdot 15$

Simplify the right side.

Multiply $- 2$ by $15$ .

$2 n + 6 \geq - 30$

Step 5

Solve for $n$ .

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

Subtract $6$ from both sides of the inequality.

$2 n \geq - 30 - 6$

Subtract $6$ from $- 30$ .

$2 n \geq - 36$

Divide each term in $2 n \geq - 36$ by $2$ and simplify.

Divide each term in $2 n \geq - 36$ by $2$ .

$\frac{2 n}{2} \geq \frac{- 36}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 n}{2} \geq \frac{- 36}{2}$

Divide $n$ by $1$ .

$n \geq \frac{- 36}{2}$

Simplify the right side.

Divide $- 36$ by $2$ .

$n \geq - 18$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$n \geq - 18$

Interval Notation:

$[- 18, \infty)$

Solve the Inequality for n (n-2)/5-n/3<=2

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