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Linear inequalities represent mathematical expressions where the two sides are not equal but have a specific relational property. They typically involve a variable where the expression might be \(\leq\) (less than or equal to) or \(\geq\) (greater than or equal to) rather than an equal sign. For example, in the problem, we use the inequality \[ x + y \leq 2000 \] to convey that the sum of student tickets and general admission tickets must not exceed 2000.
Linear inequalities form straight lines when graphed and divide the plane into two regions: one that satisfies the inequality and one that does not. Understanding how to interpret and manipulate these inequalities is a key skill in algebra.

A system of inequalities consists of multiple inequalities that need to be satisfied simultaneously. In the theater problem, we have four inequalities:
\[ x + y \leq 2000 \], which ensures the total tickets do not exceed 2000;
\[ x \geq 3y \], reserving at least three times more student tickets;
\[ x \geq 0 \] and \[ y \geq 0 \], which make sure ticket counts are non-negative.
To solve a system of inequalities, we graph each inequality on the same coordinate plane. The solution set is the region where all the inequalities overlap.

Graphing inequalities involves plotting the inequality lines and shading the appropriate regions. Here’s how to graph each inequality from the exercise:
1. \[ x + y \leq 2000 \]: Graph this as a line that intercepts the y-axis at 2000 and the x-axis at 2000. Shade below the line.
2. \[ x \geq 3y \]: This forms a line through the origin where for every 1 unit of x, y increases by 1/3. Shade above this line.
3. \[ x \geq 0 \]: Shade to the right of the y-axis.
4. \[ y \geq 0 \]: Shade above the x-axis.
The combined shaded areas from all inequalities represent the feasible region. This region is where all conditions are met.

College algebra often deals with more complex problems like systems of inequalities. These problems require skills in both algebraic manipulation and graphical representation. Students learn to:

• Define and interpret inequalities.
• Formulate equations representing real-world scenarios.
• Graph solutions on a coordinate plane.
• Find regions where multiple inequalities overlap.

In the college theater problem, applying these skills helps determine how many of each type of ticket can be sold while staying within constraints.

Non-negative constraints pertain to variables that cannot take negative values. Often, this is because negative quantities do not make sense in the context of the problem. In the theater problem, it does not make sense to sell a negative number of tickets. Therefore, we use the inequalities \[ x \geq 0 \] and \[ y \geq 0 \] to ensure the number of tickets is always zero or positive. These constraints are essential in maintaining realistic and interpretable solutions within the feasible region of a graph.