Restaurant Seating. As part of a remodeling project, a...

Restaurant Seating. As part of a remodeling project, a restaurant owner will be installing new booths that seat 4 persons, and new tables that seat 6 persons. The overall seating must conform to the sign shown below. Write an inequality that describes the possible combinations of the number of booths ( $x$ ) and the number of tables ( $y$ ) that the owner can install. Graph the inequality and give three ordered-pair solutions.

Key Concepts

Ordered Pair Solutions

Ordered pair solutions are specific pairs of values (represented as (x, y)) that satisfy the conditions of an equation or inequality. Choosing valid ordered pairs from the graph or algebraic solution demonstrates that these combinations adhere to the constraints defined by the inequality.

Linear Inequalities

A linear inequality involves a linear expression with two or more variables that uses inequality symbols (such as ?, ?, <, or >) to represent a range of possible solutions. This concept is essential for modeling constraints in real-world problems where conditions limit the values that variables can take.

Translating Word Problems into Inequalities

This concept focuses on converting a problem described in words into a mathematical inequality. It involves identifying the variables, understanding the relationships between quantities, and setting up an inequality that accurately represents the conditions or limits specified in the problem.

Graphing Linear Inequalities

Graphing a linear inequality involves first drawing the corresponding boundary line (which represents the equation where the inequality is an equality) on a coordinate plane, and then shading the region that contains all the solutions to the inequality. This visual representation helps in understanding the range of valid solutions.

Feasible Region

The feasible region is the set of all points on a graph that satisfy the given inequality or system of inequalities. It represents all the possible combinations of variable values that meet the problem’s constraints and is a key concept in optimization and decision-making problems.

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