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Chapter 3: Problem 30

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

Short Answer

Expert verified
The solution set for the given system of inequalities is the closed, bounded region on the coordinate plane, formed by the overlapping area where all four inequalities are satisfied simultaneously.

Step by step solution

01

Sketch the boundary lines

To graph inequalities, we need to begin by sketching the boundaries of the inequality. We'll first convert each inequality to an equation and then plot the lines. 1. \(3x + 4y = 12\) 2. \(2x - y = -2\) 3. \(y = 0\) and \(y = 3\) 4. \(x = 0\) In order to plot these lines, we will need to determine their slope and y-intercepts. 1. Slope = \(-\frac{3}{4}\), Y-intercept = (0, 3) 2. Slope = 2, Y-intercept = (0, 2) 3. First, the x-axis; second a horizontal line parallel to the x-axis at y=3 4. The y-axis
02

Shade the regions

Now, we can shade the regions satisfying each inequality: 1. For \(3x + 4y \geq 12\), choose a test point not on the line (0,0) and check if it satisfies the inequality. Since it doesn't (0 is not greater than or equal to 12), we shade the region opposite the test point, so above the line. 2. For \(2x - y \geq -2\), choose the same test point (0,0). It satisfies the inequality, so we shade the side of the line the test point is on, so above the line. 3. For \(0 \leq y \leq 3\), we see that two horizontal lines bound the region between the x-axis and y = 3. 4. For \(x \geq 0\), the line x = 0 is the y-axis, and we need to consider the right side of it.
03

Identify the solution set

Now we must find the overlapping region that satisfies all 4 inequalities. This region should be shaded by all 4 conditions. When we look at our graph, we can see that there is indeed an overlapping region, which indicates a solution set. We observe that this region has a bounded shape.
04

Determine if the solution set is bounded or unbounded

The overlapping region we determined in the previous step has a closed shape bounded by the lines we graphed. Therefore, the solution set is bounded.
05

Final Answer

The solution set for the system of inequalities is the closed, bounded region on the coordinate plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bounded Solution
A bounded solution in the context of graphing inequalities refers to a limited area where all the inequalities are satisfied simultaneously. This means the solution set forms a closed region within the coordinate plane.

This bounded region is surrounded by boundary lines, and it does not extend infinitely in any direction. For the given exercise, the solution set is formed by the convergence of various lines, creating a polygonal shape that limits the area of potential solutions.

Characteristics of a bounded solution:
  • The solution set does not go on indefinitely.
  • All solutions fall within a closed and confined area.
  • This area is determined by the inequalities establishing boundaries.
Identifying whether a solution set is bounded or unbounded helps understand the nature of possible solutions, whether they are limited or infinite.
System of Inequalities
A system of inequalities is a set of two or more inequalities that are considered simultaneously. The goal is to find a region on the graph that satisfies all the inequalities at the same time.

In mathematical terms, you're looking for any point within the coordinate system that makes each inequality true. In our exercise, the inequalities are:
  • \(3x + 4y \geq 12\)
  • \(2x - y \geq -2\)
  • \(0 \leq y \leq 3\)
  • \(x \geq 0\)
Each inequality represents a separate rule about how the variables behave. When these are graphed, the overlapping region that meets all conditions is the solution. Thinking of inequalities as rules helps visualize how the system works together, finding common ground for all conditions.
Graphical Solution Method
The graphical solution method involves plotting each inequality on a graph and identifying the region where they overlap. This helps visually understand and solve the system of inequalities.

The process typically involves several steps:
  • Convert each inequality into an equation to sketch boundary lines.
  • Calculate slopes and intercepts to accurately draw the lines.
  • Use test points to determine which side of the line the solution lies.
  • Shade the regions satisfying each inequality.
  • Locate where all shaded regions overlap to find the solution set.
For example, in our exercise, after plotting each line, we used (0,0) as a test point to decide where to shade for each inequality.

The ultimate goal is to highlight the closed, bounded region where all conditions meet. This visual method is intuitive and provides a clear representation of the solutions, making it especially helpful for those who are learning about systems of inequalities.

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Most popular questions from this chapter

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} x+2 y & \geq 3 \\ 5 x-4 y & \leq 16 \\ 0 \leq y & \leq 2 \\ x & \geq 0 \end{aligned} $$

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National Business Machines manufactures two models of fax machines: \(\mathrm{A}\) and \(\mathrm{B}\). Each model A costs \(\$ 100\) to make, and each model B costs \(\$ 150\). The profits are \(\$ 30\) for each model \(\mathrm{A}\) and \(\$ 40\) for each model B fax machine. If the total number of fax machines demanded per month does not exceed 2500 and the company has earmarked no more than \(\$ 600,000 /\) month for manufacturing costs, how many units of each model should National make each month in order to maximize its monthly profit? What is the optimal profit?

WATER SuPPLY The water-supply manager for a Midwest city needs to supply the city with at least 10 million gal of potable (drinkable) water per day. The supply may be drawn from the local reservoir or from a pipeline to an adjacent town. The local reservoir has a maximum daily yield of 5 million gallons of potable water, and the pipeline has a maximum daily yield of 10 million gallons. By contract, the pipeline is required to supply a minimum of 6 million gallons/day. If the cost for 1 million gallons of reservoir water is \(\$ 300\) and that for pipeline water is \(\$ 500\), how much water should the manager get from each source to minimize daily water costs for the city?
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