Learning Objectives for Graphing Systems of Linear Inequalities

A production manager is comparing two operational costs represented by the inequality ${}2a < 5a + 12$. After simplifying the inequality to ${}-3a < 12$, what is the mandatory rule for handling the inequality symbol in the final step of dividing both sides by -3?

An inventory manager is solving the inequality ${}2a < 5a + 12$ to determine reorder levels. After subtracting ${}5a$ from both sides, the manager gets $-3a < 12$. To finish solving for $a$, the manager divides both sides by -3 and concludes that the final solution is $a < -4$.

A manufacturing manager is using the linear inequality ${}2a < 5a + 12$ to calculate a safety threshold for an automated assembly line, where $a$ represents the number of units processed per minute. Arrange the following steps in the correct order to solve for $a$.

Standardizing the Solution for Cost Comparisons

In a corporate logistics scenario, a cost constraint is modeled by the linear inequality ${}2a < 5a + 12$. Match each stage of the solution process with its correct mathematical description or result.

In a corporate inventory audit, a logistics manager models safety stock level variations using the linear inequality ${}2a < 5a + 12$, where $a$ represents the number of buffer cases. After subtracting ${}5a$ from both sides, the manager simplifies the relation to ${}-3a < 12$. To isolate the variable, the manager divides both sides by -3 and recalls the rule to reverse the inequality symbol (changing it from $<$ to $>$). Fill in the blank to complete the final solution: $a >$ ____.