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Chapter 8: Problem 87

What is a linear inequality in two variables? Provide an example with your description.

Short Answer

Expert verified
A linear inequality in two variables is a relationship between two variables that can be graphically represented as a line, along with a shaded region. An example would be the inequality \(y > 2x + 3\), where the solution region is the area above the line \(y = 2x + 3\)

Step by step solution

01

Definition of Linear Inequality in Two Variables

A linear inequality in two variables is a relationship between two different variables that forms a region in the coordinate plane. This inequality can be represented on a graph as a line and a shaded region either above or below the line. The line divides the plane into two halves, and the inequality specifies which half-plain is considered the solution region. A point in this region will satisfy the inequality, whereas a point outside of it will not.
02

Example of Linear Inequality in Two Variables

For example, consider the inequality \(y > 2x + 3\). This is a linear inequality in two variables. To graph this, begin by drawing the line \(y = 2x + 3\). Since the inequality symbol is '>', this means that the solution region is above the line. So, the region above the line (excluding the line itself, since it's not \(y \geq 2x + 3\)) is shaded to indicate the solution region. Any point within this shaded region fulfills the inequality.

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Most popular questions from this chapter

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is 125 for the rear-projection televisions and 200 for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and let \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit. b. The manufacturer is bound by the following constraints: \(\cdot\) Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \(\cdot\) Equipment in the factory allows for making at most 200 plasma televisions in one month. \(\cdot\) The cost to the manufacturer per unit is 600 for the rear-projection televisions and 900 for the plasma televisions. Total monthly costs cannot exceed 360,000. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200)\) \((450,100), \text { and }(450,0) .]\) e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing _____ rear- projection televisions each month and _____ \(-\) plasma televisions each month. The maximum monthly profit is $_____.

Group members should choose a particular field of interest. Research how linear programming is used to solve problems in that field. If possible, investigate the solution of a specific practical problem. Present a report on your findings, including the contributions of George Dantzig, Narendra Karmarkar, and L. G. Khachion to linear programming.

The function $$f(t)=\frac{25.1}{1+2.7 e^{-0.05 t}}$$ models the population of Florida, \(f(t),\) in millions, \(t\) years after \(1970 .\) a. What was Florida's population in \(1970 ?\) b. According to this logistic growth model, what was Florida's population, to the nearest tenth of a million. in \(2010 ?\) Does this underestimate or overestimate the actual 2010 population of 18.8 million? c. What is the limiting size of the population of Florida?

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=4 x+2 y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+3 y \leq 12} \\ {3 x+2 y \leq 12} \\ {x+y \geq 2} \end{array}\right. \end{aligned} $$

will help you prepare for the material covered in the next section. Solve by the substitution method: $$\left\\{\begin{array}{l}{4 x+3 y=4} \\\\{y=2 x-7}\end{array}\right.$$
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