Try It: Solving $\frac{1}{x^2 + 2x - 8} > 0$

Try It: Solving $\frac{3}{x^2 + x - 12} > 0$

An inventory analyst models the surplus ratio of a component using the rational expression $\frac{120}{x^2 - 4x - 12}$, where $x$ represents the production rate. To determine the production rates that yield a strictly positive surplus, the analyst sets up the inequality $\frac{120}{x^2 - 4x - 12} > 0$.

Recalling the procedure for solving a rational inequality with a constant numerator, which of the following statements correctly describes how to find the partition numbers (critical values) for this model?

A technician monitoring a cooling system uses the rational expression $\frac{40}{w^2 - 16}$ to evaluate air pressure stability, where $w$ represents the fan speed in meters per second. To find the speeds that maintain positive stability, the technician must solve the inequality $\frac{40}{w^2 - 16} > 0$. Arrange the following steps in the correct order to solve this rational inequality with a constant numerator.

A logistics coordinator uses the rational expression $\frac{150}{w - 30}$ to calculate the 'Cargo Stability Index,' where $w$ is the total weight of the shipment in tons. To determine the weight ranges that maintain a positive stability index, the coordinator solves the inequality $\frac{150}{w - 30} > 0$.

True or False: In this inequality, there are no 'zero' partition numbers because the numerator is a non-zero constant (150).

A safety inspector at a logistics hub uses the 'Stress Impact' formula $R = \frac{100}{x^2 - 25}$ to evaluate shelving loads, where x represents the weight in tons. To ensure safety, the inspector must solve the inequality $\frac{100}{x^2 - 25} > 0$. Match each component of this process with its correct description or value based on the standard procedure for solving rational inequalities.

Analyzing Load Distribution Partition Numbers

A quality control technician at an electronics manufacturing plant models the failure rate of a sensor using the rational expression $F(t) = \frac{15}{t^2 - 6t + 8}$, where $t$ is the operating temperature in degrees Celsius. To find the temperature range where the failure rate remains positive, the technician solves the inequality $\frac{15}{t^2 - 6t + 8} > 0$.

During the solution process, the technician recalls that because the numerator is a non-zero constant (15), there are no partition numbers where the expression equals zero. Therefore, any partition numbers for this inequality can only occur where the expression is ____.

Troubleshooting a Supply Chain Efficiency Model