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.
$$
\begin{aligned}
6 x+7 y & \leq 84 \\
12 x-11 y & \leq 18 \\
6 x-7 y & \leq 28 \\
x \geq 0, y & \geq 0
\end{aligned}
$$

Key Concepts

System of Linear Inequalities

A system of linear inequalities consists of several inequality statements, each representing a half-plane in a coordinate system. When combined, these inequalities specify a set of possible solutions that satisfy all constraints simultaneously. This forms the basis for defining a feasible region, which is crucial in optimization and linear programming problems.

Feasible Region

The feasible region is the set of all points that satisfy all the given inequalities simultaneously. Graphically, this region is found by plotting each inequality as a half-plane and then identifying the area where all half-planes overlap. This concept is key as it represents the solution set for the system.

Graphical Representation of Inequalities

Graphically representing inequalities involves drawing their corresponding boundary lines (by converting the inequality to an equation) and then determining which side of the line represents the inequality using a test point. This method helps in visually identifying the feasible region defined by the system.

Bounded vs. Unbounded Regions

A region is considered bounded if it is enclosed completely by the inequality boundaries, meaning that it can be contained within a finite area. Conversely, an unbounded region extends infinitely in one or more directions. Determining whether the feasible region is bounded or unbounded is important in optimization since bounded regions often imply the existence of optimal solutions in problems like linear programming.

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