Write a system of inequalities that describes the possible...

Write a system of inequalities that describes the possible solutions to each problem and graph the solution set to the system. Delicate Balance A fast food restaurant must have a minimum of 30 employees and a maximum of $50 .$ To avoid charges of sexual bias, the company has a policy that the number of employees of one sex must never exceed the number of employees of the other sex by more than six. How many persons of each sex could be employed at this restaurant?

Write a system of inequalities that describes the possible solutions to each problem and graph the solution set to the system.
Delicate Balance A fast food restaurant must have a minimum of 30 employees and a maximum of $50 .$ To avoid charges of sexual bias, the company has a policy that the number of employees of one sex must never exceed the number of employees of the other sex by more than six. How many persons of each sex could be employed at this restaurant?

Key Concepts

Translation of Word Problems to Mathematical Models

This key concept involves carefully reading and analyzing a real-world scenario to extract the underlying mathematical relationships and constraints. The process includes identifying quantities, understanding the conditions that they satisfy, and then writing those conditions in the form of algebraic expressions or inequalities. This skill bridges the gap between practical problems and mathematical analysis.

Absolute Value Inequalities

Absolute value inequalities are used to express constraints involving the magnitude of differences between quantities, regardless of their sign. They are particularly useful when a problem states that one value should not exceed another by more than a specified amount, capturing the idea of a maximum allowable difference. Solving these inequalities often involves rewriting them as a pair of inequalities without the absolute value notation.

System of Inequalities

A system of inequalities consists of multiple inequalities that are considered simultaneously. The solution to the system is the set of values that satisfies all of the inequalities at once. This is a foundational concept in algebra used to express constraints and conditions imposed by real-world problems and is closely related to the idea of feasible regions in optimization problems.

Graphical Representation of Inequalities

Graphing inequalities involves depicting the set of all possible solutions on a coordinate plane. Each inequality defines a region (often a half-plane) bounded by a line, where the line itself may or may not be included in the solution set. The overall solution to a system is found by shading the regions for each inequality and then identifying the intersection where all conditions are met.

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