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Chapter 3: Problem 39

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 a system of linear inequalities in two variables is bounded if it can be enclosed by a rectangle.

Short Answer

Expert verified
The given statement is true. The solution set of a system of linear inequalities in two variables is bounded if it can be enclosed by a rectangle because any solution set that can be enclosed by a rectangle will have at least two linear inequalities defining the horizontal boundaries and two defining the vertical boundaries, ensuring that the solution set is bounded within a finite region of the plane.

Step by step solution

01

Evaluating the given statement

Let's consider the given statement and try to interpret it in terms of linear inequalities: "The solution set of a system of linear inequalities in two variables is bounded if it can be enclosed by a rectangle." We can hypothesize that if the solution set can be enclosed by a rectangle, there are at least two linear inequalities defining the horizontal boundaries (top and bottom) and two defining the vertical boundaries (left and right).
02

Testing the statement with an example

Let's test the given statement with an example of a system of linear inequalities that can be enclosed by a rectangle: Inequality 1: \( y \leq x + 2 \) (top boundary) Inequality 2: \( y \geq x - 2 \) (bottom boundary) Inequality 3: \( y \leq -x + 4 \) (right boundary) Inequality 4: \( y \geq -x \) (left boundary) It is easy to see that the solution set satisfies all four inequalities and forms a parallelogram, which can be enclosed by a rectangle. In this case, the statement appears to be true.
03

Verifying the statement for a general case

However, we need to verify if the statement holds for any system of linear inequalities that can be enclosed by a rectangle. It can be proved that if there exists a rectangle enclosing the solution set for a system of linear inequalities, the solution set will always be bounded. Therefore, based on our analysis, the given statement is indeed true for any system of linear inequalities that can be enclosed by a rectangle. So, the statement is true.
04

Conclusion

In conclusion, the statement "The solution set of a system of linear inequalities in two variables is bounded if it can be enclosed by a rectangle" is true. This is because any solution set that can be enclosed by a rectangle will have at least two linear inequalities defining the horizontal boundaries and two defining the vertical boundaries, which ensure that the solution set is bounded within a finite region of the plane.

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Most popular questions from this chapter

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 gal 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? What is the minimum daily cost?

A farmer plans to plant two crops, A and B. The cost of cultivating crop A is \(\$ 40\) /acre whereas that of crop \(\mathrm{B}\) is \(\$ 60\) /acre. The farmer has a maximum of \(\$ 7400\) available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop \(\underline{B}\) requires 25 labor-hours. The farmer has a maximum of 3300 labor-hours available. If she expects to make a profit of \(\$ 150\) /acre on crop \(\mathrm{A}\) and \(\$ 200\) acre on crop \(\mathrm{B}\), how many acres of each crop should she plant in order to maximize her profit? What is the optimal profit?

MaNUFACTURING-PRODUCTION SCHEDULING A company manufactures products \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Each product is processed in three departments: I, II, and III. The total available labor-hours per week for departments I, II, and III are 900,1080, and 840, respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows: $$ \begin{array}{lccc} \hline & \begin{array}{c} \text { Product } \\ \mathbf{A} \end{array} & \begin{array}{c} \text { Product } \\ \mathbf{B} \end{array} & \begin{array}{c} \text { Product } \\ \mathbf{C} \end{array} \\ \hline \text { Dept. I } & 2 & 1 & 2 \\ \hline \text { Dept. II } & 3 & 1 & 2 \\ \hline \text { Dept. III } & 2 & 2 & 1 \\ \hline \text { Profit } & \$ 18 & \$ 12 & \$ 15 \\ \hline \end{array} $$ How many units of each product should the company produce in order to maximize its profit?

Solve each linear programming problem by the method of corners. $$ \begin{array}{ll} \text { Maximize } & P=x+2 y \\ \text { subject to } & x+y \leq 4 \\ & 2 x+y \leq 5 \\ x & \geq 0, y \geq 0 \end{array} $$

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{r} x+y \leq 4 \\ 2 x+y \leq 6 \\ 2 x-y \geq-1 \\ x \geq 0, y \geq 0 \end{array} $$
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