The gymnasium at a local high school has 1000 seats. The...

The gymnasium at a local high school has 1000 seats. The state basketball championship game will be held there next Friday night. Tickets to the game will be priced as follows: $\$ 10$ if purchased in advance and $\$ 15$ if purchased at the door. The booster club feels that at least 100 tickets will be sold at the advance ticket price. Total sales of at least $\$ 5000$ are expected. (a) Use $x$ to denote the number of tickets sold in advance. Use $y$ to denote the number of tickets sold at the door. Write a system of linear inequalities that describes the possible numbers of tickets sold at each price. (b) Graph the region that represents all possible combinations of ticket sales.

The gymnasium at a local high school has 1000 seats. The state basketball championship game will be held there next Friday night. Tickets to the game will be priced as follows: $\$ 10$ if purchased in advance and $\$ 15$ if purchased at the door. The booster club feels that at least 100 tickets will be sold at the advance ticket price. Total sales of at least $\$ 5000$ are expected.
(a) Use $x$ to denote the number of tickets sold in advance. Use $y$ to denote the number of tickets sold at the door. Write a system of linear inequalities that describes the possible numbers of tickets sold at each price.
(b) Graph the region that represents all possible combinations of ticket sales.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line of the inequality on a coordinate plane and then shading the region that represents the set of solutions. This visualization helps in understanding the overlap of multiple constraints in a system, highlighting the feasible region where all conditions are met.

Feasible Region

The feasible region is the set of all possible points that satisfy all of the constraints in a system of inequalities. It represents all allowable solutions to the problem and is often visualized graphically, typically as a bounded or unbounded area on the coordinate plane.

Systems of Linear Inequalities

A system of linear inequalities consists of two or more linear inequalities that are considered simultaneously. The solution to such a system is the set of all points that satisfy all the inequalities at the same time, representing intersecting constraints in many applied problems.

Linear Inequalities

Linear inequalities are mathematical expressions that relate linear functions using inequality signs (>, <, ?, ?) instead of an equals sign. They are used to represent ranges of values and are foundational in formulating constraints in many real-world scenarios.

Constraints in Word Problems

In word problems, constraints are the conditions or limits imposed by real-life situations, such as capacity or budget limits. These conditions are converted into mathematical inequalities that accurately model the scenario and guide the search for feasible solutions.

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