Try It: Solving $\frac{1}{2} + \frac{4}{x^2} < \frac{3}{x}$

Try It: Solving $\frac{1}{3} + \frac{6}{x^2} < \frac{3}{x}$

A logistics supervisor is evaluating a distribution model that uses the rational inequality $\frac{1}{3} - \frac{2}{x^2} < \frac{5}{3x}$. To find the valid range for the variable $x$, the supervisor must follow a specific sequence of algebraic steps. Place the following actions in the correct order according to the standard procedure for solving such inequalities.

An industrial technician is using the rational inequality $\frac{1}{3} - \frac{2}{x^2} < \frac{5}{3x}$ to model the efficiency of a cooling system. To solve this inequality, the first major step after moving all terms to one side is to combine them using a common denominator. Based on the provided example, what is the Least Common Denominator (LCD) used for this calculation?

An industrial technician is evaluating a crane's load safety using the rational inequality $\frac{1}{3} - \frac{2}{x^2} < \frac{5}{3x}$. After combining the terms into a single rational expression, the technician simplifies the inequality to $\frac{(x - 6)(x + 1)}{3x^2} < 0$. To solve this, the technician identifies the partition numbers that divide the number line into intervals. If two of the partition numbers are -1 and 6, what is the third partition number? ____

A systems analyst is optimizing a server's response time using the rational inequality $\frac{1}{3} - \frac{2}{x^2} < \frac{5}{3x}$. Match the following components of the solution process to their correct roles or values based on the simplified expression $\frac{(x - 6)(x + 1)}{3x^2} < 0$.

A supply chain manager is modeling warehouse efficiency with a rational inequality that simplifies to $\frac{(x - 6)(x + 1)}{3x^2} < 0$. True or False: When determining the final valid intervals for this model, the partition number derived from setting the denominator to zero ($x = 0$) must be included in the solution set.