Problem 34 - 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. $$ \begin{aligned} 6 x+7 y & \leq 84 \\ 12 x-11 y & \leq 18 \\ 6 x-7 y & \leq 28 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Step-by-Step Solution

Answer

The solution set for the system of inequalities is a bounded region in the first quadrant where all inequalities are true. This region is enclosed by the lines \(y \leq -\frac{6}{7}x + 12\), \(y \leq \frac{11}{12}x - \frac{3}{2}\), \(y \geq \frac{6}{7}x - 4\), and \(y \geq 0\).

Step 1: Convert inequalities to slope-intercept form

Rewrite each inequality in terms of y: \[ \begin{aligned} 7y &\leq -6x + 84 &\Rightarrow y &\leq -\frac{6}{7}x + 12 \\ 11y &\geq 12x - 18 &\Rightarrow y &\leq \frac{11}{12}x - \frac{3}{2} \\ 7y &\geq 6x - 28 &\Rightarrow y &\geq \frac{6}{7}x - 4 \\ \end{aligned} \] The fourth inequality, \(y \geq 0\), is already in slope-intercept form.

Step 2: Plot the inequalities on the coordinate plane

Sketch each of the inequality lines on the coordinate plane: 1. Line 1: \(y = -\frac{6}{7}x + 12\). Plot the points \((0, 12)\) and \(\left(7,6\right)\) and draw a solid line through them, since the inequality sign is \(\leq\). 2. Line 2: \(y = \frac{11}{12}x - \frac{3}{2}\). Plot the points \(\left(0,-\frac{3}{2}\right)\) and \((2, 1)\) and draw a solid line through them, since the inequality sign is \(\leq\). 3. Line 3: \(y = \frac{6}{7}x - 4\). Plot the points \((0, -4)\) and \((7, 2)\) and draw a solid line through them, since the inequality sign is \(\geq\). 4. Plot the \(y \geq 0\) inequality by drawing a solid line along the x-axis. For each line, shade the area that satisfies the inequality. The region where all shaded areas overlap is the solution set.

Step 3: Identify the solution set and determine if it is bounded or unbounded

By examining the plot, we can see that the area where all shaded regions overlap is a polygon located in the first quadrant. Since this polygon is enclosed by lines, the solution set is bounded. The final solution is a bounded region in the first quadrant where all inequalities are true.

Key Concepts

System of Inequalities

Understanding a system of inequalities is essential when dealing with multiple conditions that have to be met simultaneously. A system of inequalities consists of two or more inequalities that have to be satisfied at the same time by the same set of variables. In the given exercise, we are looking at a system that contains four inequalities involving two variables, x and y. To solve it, we graph each inequality on the same coordinate plane and look for the region that satisfies all conditions at once.

Often, these systems represent constraints in real-world scenarios like budget limits, space requirements, or other optimization problems. The intersection of the various solution sets of each individual inequality gives us the set solutions of the system. It can either be a specific region (bounded) or all the area beyond a certain line (unbounded), depending on the inequalities' nature.

Slope-Intercept Form

The slope-intercept form is a straightforward way to write linear equations. It's given by the formula y = mx + b, where m is the slope of the line, and b is the y-intercept, the point at which the line crosses the y-axis. This form is particularly useful for graphing since it allows us to draw the line quickly by identifying the slope and the y-intercept.

In the exercise, the inequalities are converted to the slope-intercept form. This conversion facilitates plotting as you can readily identify the line's direction (positive or negative slope) and starting point on the graph (the y-intercept). With these two pieces of information, graphing the inequalities becomes a task that is both manageable and accurate.

Coordinate Plane Plotting

Plotting inequalities on a coordinate plane may seem challenging at first, but with practice, it becomes an engaging way to visualize math problems. After converting inequalities to the slope-intercept form, plotting involves drawing lines for each equation and then shading the appropriate side of these lines that represent the solution for each inequality. By identifying key points such as intercepts and using the slope, you create an accurate representation of each inequality.

It’s important to note that a solid line is used when the inequality is either ≤ or ≥, indicating that points on the line are included in the solution, whereas a dashed line is used for < or >, showing excluded points. After plotting all lines, the intersection area, where all shaded parts meet, is the solution to the system.

Bounded and Unbounded Regions

In relation to systems of inequalities, bounded and unbounded regions are concepts that describe the nature of the solution set on a coordinate plane. A region is considered bounded if it's enclosed by lines and has a finite area; this often looks like a polygon. Conversely, an unbounded region refers to an area that extends infinitely in one or more directions, not enclosed by the lines.

The exercise provides a perfect illustration of a bounded region, as the overlapping shaded areas form a close-ended shape in the first quadrant. Recognizing whether a solution set is bounded is important in understanding the constraints of the problem and can have practical implications for how the solutions are implemented in real-world contexts.