Question

solve the rational inequality. write your answer in interval notation.

\\\frac{4}{c-2} - \frac{3}{c} < 0\\

Explanation:

Step1: Combine terms

Find a common denominator to combine the fractions.

$$\frac{4c - 3(c - 2)}{c(c - 2)} < 0$$

Step2: Simplify numerator

Simplify the expression in the numerator.

$$\frac{c + 6}{c(c - 2)} < 0$$

Step3: Find boundary points

Identify the critical points where the expression is zero or undefined.

$$c = -6, \quad c = 0, \quad c = 2$$

Step4: Test intervals

Determine the sign of the expression in each interval.

$$\text{For } c \in (-\infty, -6), \quad \frac{-}{(-)(-)} < 0 \quad (\text{True})$$

$$\text{For } c \in (-6, 0), \quad \frac{+}{(-)(-)} > 0 \quad (\text{False})$$

$$\text{For } c \in (0, 2), \quad \frac{+}{(+)(-)} < 0 \quad (\text{True})$$

$$\text{For } c \in (2, \infty), \quad \frac{+}{(+)(+)} > 0 \quad (\text{False})$$

Answer:

\((-\infty, -6) \cup (0, 2)\)