Determine whether the statement is true or false. If it is...

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The solution set of the system $$ \begin{array}{l} a x+b y \leq e \\ c x+d y \leq f \\ x \geq 0, y \geq 0 \end{array} $$ where $a, b, c, d, e$, and $f$ are positive real numbers, is a bounded set.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
The solution set of the system
$$
\begin{array}{l}
a x+b y \leq e \\
c x+d y \leq f \\
x \geq 0, y \geq 0
\end{array}
$$
where $a, b, c, d, e$, and $f$ are positive real numbers, is a bounded set.

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Key Concepts

Linear Inequalities

Linear inequalities are statements that relate linear expressions using inequality symbols. They define half-planes in the coordinate system, and, when considered as a system, each inequality restricts the region where the solutions can lie.

Feasible Region

The feasible region is the intersection of all the half-planes defined by a system of inequalities. In this context, it is the set of all points (x, y) that satisfy each inequality simultaneously, including the nonnegative constraints x ? 0 and y ? 0.

Boundedness

Boundedness refers to whether a set is contained within some finite region of space. In systems of inequalities, the feasible region is bounded if it does not extend infinitely in any direction. Here, given that all coefficients in the inequalities are positive and the region is further restricted to the first quadrant by x ? 0 and y ? 0, the solution set is confined by the intercepts of the lines with the axes, ensuring that it lies within a finite area.

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