To solve a rational inequality that requires combining fractions, such as $\frac{1}{3} - \frac{2}{x^2} < \frac{5}{3x}$, first rewrite it so that $0$ is on one side. Subtracting $\frac{5}{3x}$ yields $\frac{1}{3} - \frac{2}{x^2} - \frac{5}{3x} < 0$. Next, rewrite each fraction using the least common denominator (LCD), which is $3x^2$, to combine them into a single rational expression: $\frac{x^2 - 5x - 6}{3x^2} < 0$. Then, factor the numerator to obtain $\frac{(x - 6)(x + 1)}{3x^2} < 0$. After factoring, determine the zero partition numbers by finding the values where the numerator or denominator equals zero. Setting the numerator factors to zero gives $x = 6$ and $x = -1$, while setting the denominator to zero gives $x = 0$. Use these partition numbers to divide the number line into intervals: $(-\infty, -1)$, $(-1, 0)$, $(0, 6)$, and $(6, \infty)$. By evaluating the sign of each factor in these intervals, determine the overall sign of the quotient. The quotient is negative in the intervals $(-1, 0)$ and $(0, 6)$. Since the inequality is strictly less than zero (<$), the partition numbers that make the expression equal to zero are excluded. Because zero partition numbers from the denominator are always excluded, the solution in interval notation is the union of these two intervals: (-1, 0) \cup (0, 6)$$.