Determine graphically the solution set for each system of...

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{array}{l}
4 x-3 y \leq 12 \\
5 x+2 y \leq 10 \\
x \geq 0, y \geq 0
\end{array}
$$

Key Concepts

Systems of Linear Inequalities

This concept deals with multiple linear inequalities considered simultaneously. The solution to a system is found by identifying the common region (or feasible set) that satisfies all of the individual inequalities. This process is foundational in areas such as linear programming and optimization.

Graphical Representation of Linear Inequalities

This concept involves plotting the boundary line of each inequality, which is derived by replacing the inequality sign with an equality, on the coordinate plane. Once these lines are drawn, the plane is divided into two half?planes, and the appropriate side (or half?plane) corresponding to the inequality is determined through testing a point not on the line. This visualization helps in understanding the solution set of each inequality.

Half-Planes

A half-plane is one of the two regions into which a straight line divides the coordinate plane. Each linear inequality corresponds to one of these half-planes by indicating whether the region on one side of the boundary line (the line derived from the equality) satisfies the inequality. Recognizing half-planes is essential for correctly shading the feasible region for a system of inequalities.

Feasible Region

The feasible region refers to the set of all points that satisfy every inequality in a given system simultaneously. It is the intersection of the half-planes produced by each inequality. Identifying this region is a crucial step in solving systems of linear inequalities, particularly when the solution is presented graphically.

Bounded Versus Unbounded Regions

A bounded region is enclosed within a closed perimeter, meaning that it does not extend infinitely in any direction, while an unbounded region extends without limit along at least one direction. Determining whether a feasible region is bounded or unbounded is important in optimization problems and understanding the nature of the solution set in systems of inequalities.

Please give Ace some feedback