Solve the Inequality for q 3/4q-2/3q<5/12q-1/12

$\frac{3}{4} q - \frac{2}{3} q < \frac{5}{12} q - \frac{1}{12}$

Step 1

Simplify $\frac{3}{4} q - \frac{2}{3} q$ .

Simplify each term.

Combine $\frac{3}{4}$ and $q$ .

$\frac{3 q}{4} - \frac{2}{3} q < \frac{5}{12} q - \frac{1}{12}$

Combine $q$ and $\frac{2}{3}$ .

$\frac{3 q}{4} - \frac{q \cdot 2}{3} < \frac{5}{12} q - \frac{1}{12}$

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

$\frac{3 q}{4} - \frac{2 q}{3} < \frac{5}{12} q - \frac{1}{12}$

To write $\frac{3 q}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 q}{4} \cdot \frac{3}{3} - \frac{2 q}{3} < \frac{5}{12} q - \frac{1}{12}$

To write $- \frac{2 q}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 q}{4} \cdot \frac{3}{3} - \frac{2 q}{3} \cdot \frac{4}{4} < \frac{5}{12} q - \frac{1}{12}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 q}{4}$ by $\frac{3}{3}$ .

$\frac{3 q \cdot 3}{4 \cdot 3} - \frac{2 q}{3} \cdot \frac{4}{4} < \frac{5}{12} q - \frac{1}{12}$

Multiply $4$ by $3$ .

$\frac{3 q \cdot 3}{12} - \frac{2 q}{3} \cdot \frac{4}{4} < \frac{5}{12} q - \frac{1}{12}$

Multiply $\frac{2 q}{3}$ by $\frac{4}{4}$ .

$\frac{3 q \cdot 3}{12} - \frac{2 q \cdot 4}{3 \cdot 4} < \frac{5}{12} q - \frac{1}{12}$

Multiply $3$ by $4$ .

$\frac{3 q \cdot 3}{12} - \frac{2 q \cdot 4}{12} < \frac{5}{12} q - \frac{1}{12}$

Combine the numerators over the common denominator.

$\frac{3 q \cdot 3 - 2 q \cdot 4}{12} < \frac{5}{12} q - \frac{1}{12}$

Simplify the numerator.

Factor $q$ out of $3 q \cdot 3 - 2 q \cdot 4$ .

Factor $q$ out of $3 q \cdot 3$ .

$\frac{q (3 \cdot 3) - 2 q \cdot 4}{12} < \frac{5}{12} q - \frac{1}{12}$

Factor $q$ out of $- 2 q \cdot 4$ .

$\frac{q (3 \cdot 3) + q (- 2 \cdot 4)}{12} < \frac{5}{12} q - \frac{1}{12}$

Factor $q$ out of $q (3 \cdot 3) + q (- 2 \cdot 4)$ .

$\frac{q (3 \cdot 3 - 2 \cdot 4)}{12} < \frac{5}{12} q - \frac{1}{12}$

Multiply $3$ by $3$ .

$\frac{q (9 - 2 \cdot 4)}{12} < \frac{5}{12} q - \frac{1}{12}$

Multiply $- 2$ by $4$ .

$\frac{q (9 - 8)}{12} < \frac{5}{12} q - \frac{1}{12}$

Subtract $8$ from $9$ .

$\frac{q \cdot 1}{12} < \frac{5}{12} q - \frac{1}{12}$

Multiply $q$ by $1$ .

$\frac{q}{12} < \frac{5}{12} q - \frac{1}{12}$

Step 2

Combine $\frac{5}{12}$ and $q$ .

$\frac{q}{12} < \frac{5 q}{12} - \frac{1}{12}$

Step 3

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

Subtract $\frac{5 q}{12}$ from both sides of the inequality.

$\frac{q}{12} - \frac{5 q}{12} < - \frac{1}{12}$

Combine the numerators over the common denominator.

$\frac{q - 5 q}{12} < - \frac{1}{12}$

Subtract $5 q$ from $q$ .

$\frac{- 4 q}{12} < - \frac{1}{12}$

Cancel the common factor of $- 4$ and $12$ .

Factor $4$ out of $- 4 q$ .

$\frac{4 (- q)}{12} < - \frac{1}{12}$

Cancel the common factors.

Factor $4$ out of $12$ .

$\frac{4 (- q)}{4 (3)} < - \frac{1}{12}$

Cancel the common factor.

$\frac{4 (- q)}{4 \cdot 3} < - \frac{1}{12}$

Rewrite the expression.

$\frac{- q}{3} < - \frac{1}{12}$

Move the negative in front of the fraction.

$- \frac{q}{3} < - \frac{1}{12}$

Step 4

Divide each term in $- \frac{q}{3} < - \frac{1}{12}$ by $- 1$ and simplify.

Divide each term in $- \frac{q}{3} < - \frac{1}{12}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{q}{3}}{- 1} > \frac{- \frac{1}{12}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{q}{3}}{1} > \frac{- \frac{1}{12}}{- 1}$

Divide $\frac{q}{3}$ by $1$ .

$\frac{q}{3} > \frac{- \frac{1}{12}}{- 1}$

Simplify the right side.

Dividing two negative values results in a positive value.

$\frac{q}{3} > \frac{\frac{1}{12}}{1}$

Divide $\frac{1}{12}$ by $1$ .

$\frac{q}{3} > \frac{1}{12}$

Step 5

Multiply both sides by $3$ .

$\frac{q}{3} \cdot 3 > \frac{1}{12} \cdot 3$

Step 6

Simplify.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{q}{3} \cdot 3 > \frac{1}{12} \cdot 3$

Rewrite the expression.

$q > \frac{1}{12} \cdot 3$

Simplify the right side.

Cancel the common factor of $3$ .

Factor $3$ out of $12$ .

$q > \frac{1}{3 (4)} \cdot 3$

Cancel the common factor.

$q > \frac{1}{3 \cdot 4} \cdot 3$

Rewrite the expression.

$q > \frac{1}{4}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$q > \frac{1}{4}$

Interval Notation:

$(\frac{1}{4}, \infty)$

Solve the Inequality for q 3/4q-2/3q<5/12q-1/12

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