91Ó°ÊÓ

Fill in the blanks. The process of writing a rational expression as the sum or difference of two or more simpler rational expressions is called ________ ________ ________.

Fill in the blanks. A set of two or more equations in two or more variables is called a ________ of ________.

Fill in the blanks. A ________ of a system of inequalities in \( x \) and \( y \) is a point \( (x, y) \) that satisfies each inequality in the system.

In Exercises 13-16, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints. Objective function: \( z = 3x + 2y \) Constraints: \( \hspace{1cm} x \ge 0 \) \( \hspace{1cm} y \ge 0 \) \( 5x + 2y \le 20 \) \( 5x + y \ge 10 \)

In Exercises 21-24, find the minimum and maximum values of the objective function and where they occur, subject to the constraints \( x \ge 0, y \ge 0, 3x + y \le 15 \), and \( 4x + 3y \le 30 \) \( z = 5x + y \)

A merchant plans to sell two models of MP3 players at prices of \(\$ 225\) and \(\$ 250\) . The \(\$ 225\) model yields a profit of \(\$ 30\) per unit and the \(\$ 250\) model yields a profit of \(\$ 31\) per unit. The merchant estimates that the total monthly demand will not exceed 275 units. The merchant does not want to invest more than \(\$ 63,000\) in inventory for these products. What is the optimal inventory level for each model? What is the optimal profit?

An accounting firm has 780 hours of staff time and 272 hours of reviewing time available each week. The firm charges \( \$1600 \) for an audit and \( \$250 \) for a tax return. Each audit requires 60 hours of staff time and 16 hours of review time. Each tax return requires 10 hours of staff time and 4 hours of review time. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue?

In Exercises 37-40, determine whether each ordered pair is a solution of the system of linear inequalities. \( \left\\{\begin{array}{l} x^2 + y^2 \ge 36\\\ -3x + y \le 10\\\ \dfrac{2}{3}x - y \ge 5\end{array}\right. \) (a) \( (-1, 7) \) (b) \( (-5, 1) \) (c) \( (6, 0) \) (d) \( (4, -8) \)

An investor has up to \(\$ 450,000\) to invest in two types of investments. Type A pays 6\(\%\) annually and type \(B\) pays 10\(\%\) annually. To have a well- balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at least one-fourth of the portfolio is to be allocated to type \(B\) investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return?

In Exercises 45-47, determine whether the statement is true or false. Justify your answer. When solving a linear programming problem, if the objective function has a maximum value at more than one vertex, you can assume that there are an infinite number of points that will produce the maximum value.