Problem 30 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

The solution set is the region in the first quadrant of the coordinate plane where all shaded regions overlap, which is formed by the intersection of the axes and the lines \(y = -\frac{3}{4}x + 3\) and \(y = 2x + 2\). The solution set is bounded.

Step 1: Graph each inequality

To graph the inequalities, we must first rewrite them in slope-intercept form: 1. \(3x + 4y \geq 12\) becomes \(y \geq -\frac{3}{4}x + 3\) 2. \(2x - y \geq -2\) becomes \(y \leq 2x + 2\) 3. \(0 \leq y \leq 3\) 4. \(x \geq 0\) Next, draw the lines for each inequality: 1. Solid line: \(y = -\frac{3}{4}x + 3\) 2. Solid line: \(y = 2x + 2\) 3. Horizontal lines: \(y = 0\) and \(y = 3\) 4. Vertical line: \(x = 0\) Shade the regions which satisfy corresponding inequalities.

Step 2: Determine the region satisfying all inequalities

To determine the regions that satisfy the inequalities, observe where the shaded regions overlap. This area shows where all the inequalities are satisfied simultaneously.

Step 3: Determine whether the solution set is bounded or unbounded

In this case, the solution set is in the first quadrant of a coordinate plane where the axes and the two lines intersect and form a closed polygon. It is constrained on all sides and not extending towards infinity. Therefore, the solution set is bounded. In conclusion, the solution set is the region in the first quadrant of the coordinate plane where all shaded regions overlap. The solution set is bounded.

Key Concepts

Solution Set

In the world of inequalities, the term "solution set" refers to all possible points or values that satisfy a given system of inequalities. For graphic solutions, this is visually represented as a shaded region on a graph. To identify the solution set of inequalities, one must determine where all the shaded areas from different inequalities on a graph intersect. This overlapping region is the solution set because it fulfills all conditions of the inequalities simultaneously. It's like finding a common area on a Venn diagram that satisfies all the circles' conditions.

Bounded and Unbounded Regions

These terms describe the nature of the solution set in relation to the coordinate plane. A 'bounded' region is surrounded by borders that prevent it from extending to infinity. Imagine a fence enclosing a garden; it has a well-defined area. In contrast, an 'unbounded' region extends infinitely in one or more directions, like a field without a fence. When all sides of the shaded region in a graph adhere to lines or the axes and do not stretch out endlessly in any direction, the solution set is bounded. This means it has clear limitations, resembling a polygon with defined edges.

Graphical Method

The graphical method gives a visual representation of solving inequalities, especially in two variables. It involves drawing each inequality on the same coordinate plane and observing where these graphs overlay. Here's the step-by-step approach:

• Convert inequalities into equations by replacing inequality signs with equal signs.
• Draw these lines on the coordinate plane, using solid lines for inclusive inequalities (\(\geq, \leq\)) and dashed lines for non-inclusive (\(> , <\)).
• Shade the region of the plane that satisfies each inequality.

The intersection of these shaded regions reveals the solution set, providing a clear visual answer that might be less apparent through algebra alone.

Inequalities in Two Variables

Inequalities in two variables involve two different quantities that interact within constraints set by inequality symbols like \(\leq, \geq, <, >\). When graphed, they create a region of possible solutions on a two-dimensional plane. For instance, in the inequality \(y \geq -\frac{3}{4}x + 3\), \(x\) and \(y\) are the two variables. The inequality suggests that any point above or on the line \(y = -\frac{3}{4}x + 3\) is part of the solution set. This concept helps us understand real-world scenarios where multiple factors interact under certain limits or boundaries, providing a broader perspective on potential outcomes.