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. Political Correctness A political party is selling 50 dollars tickets and 100 dollars tickets for a fund-raising banquet in a hall that cannot hold more than 500 people. To show that the party represents the people, the number of 100 dollars tickets must not be greater than $20 \%$ of the total number of tickets sold. How many tickets of each type can be sold?

Write a system of inequalities that describes the possible solutions to each problem and graph the solution set to the system.
Political Correctness A political party is selling 50 dollars tickets and 100 dollars tickets for a fund-raising banquet in a hall that cannot hold more than 500 people. To show that the party represents the people, the number of 100 dollars tickets must not be greater than $20 \%$ of the total number of tickets sold. How many tickets of each type can be sold?

Key Concepts

System of Linear Inequalities

This involves writing multiple inequalities that collectively describe the constraints of a problem. Instead of a single equation, each inequality represents a condition that narrows down the range of possible solutions, and together they define all the permissible values for the variables involved.

Feasible Region

The feasible region is the set of all points that satisfy all the inequalities in the system simultaneously. In a graphical representation, this region is typically found by shading the areas that meet each inequality, and then identifying the intersection of these areas.

Constraint Modeling

This key concept refers to the process of translating real-life conditions or restrictions into mathematical inequalities. For example, capacity limitations and proportional relationships between quantities are expressed as inequalities, which then form the constraints of the system.

Graphical Representation of Inequalities

Graphing inequalities involves plotting the boundary lines corresponding to the equality part of each inequality and then shading the appropriate region that meets the inequality's condition. The overlapping shaded area of all inequalities represents the solution set for the system.

Percentage Constraint Interpretation

Interpreting a percentage constraint in a problem requires converting a percentage condition into an inequality. For instance, ensuring that one category does not exceed a certain percent of the total can be expressed algebraically by relating the part to the whole through a fractional representation.

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