A quality control technician uses the stability formula $S = \frac{3}{x^2 + x - 12}$ to monitor a chemical reaction. To determine the conditions where the reaction remains stable ($S > 0$), the technician must first identify the partition numbers where the formula is undefined. Which of the following sets of values represents these partition numbers?

A business analyst is calculating viable production levels for a new product. To ensure a positive profit margin, the production metric must satisfy the rational inequality $\frac{3}{x^2 + x - 12} > 0$. Arrange the standard steps for solving this algebraic model in the correct procedural order.

A systems analyst is using the inequality $\frac{3}{x^2 + x - 12} > 0$ to define the operating limits of a server. True or False: Because the numerator is the constant ${}3$, there are no values of $x$ for which the quotient is exactly equal to ${}0$.

A safety engineer is using the rational inequality $\frac{3}{x^2 + x - 12} > 0$ to define the stable operating range of a high-pressure cooling system. To finalize the safety report, the engineer must identify the key algebraic components of this model. Match each component of the solution process with its corresponding mathematical expression or value.

An environmental consultant uses the inequality $\frac{3}{x^2 + x - 12} > 0$ to determine safe soil contamination levels. To map out risk zones, the consultant factors the denominator into $(x + 4)(x - 3)$ to identify where the model is undefined. These values, called partition numbers, divide the number line into test intervals. If the first partition number is -4, the second partition number is ____.

Determining Safety Thresholds using Partition Numbers

Recalling Partition Number Criteria for Stability Analysis