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Chapter 5: Problem 90

What does a solid line mean in the graph of an inequality?

Short Answer

Expert verified
A solid line in the graph of an inequality indicates that the inequality is a 'less than or equal to' or a 'greater than or equal to' inequality. It means that the points lying on the line as well are part of the solution to the inequality.

Step by step solution

01

Understand the Graphical Representation of Inequalities

In a graph, inequalities are usually represented with lines. The nature of these lines, whether they are solid or dashed, is not random but holds certain meaning in terms of the inequality it represents. Inequalities can be classified as either strict (less than, greater than) or non-strict (less than or equal to, greater than or equal to).
02

Define the Meaning of a Solid Line

A solid line represents a set of points which fulfills the equation of the line. This is used when graphing non-strict inequalities. The solid line suggests that the points on that line are included in the solution of the inequality. In other words, if you substitute the coordinates of any point on the solid line into the inequality, it would make the inequality true.
03

Interpret the Solid Line in an Inequality

So, when a solid line is used in the graph of an inequality, it implies that the inequality is either 'greater than or equal to' or 'less than or equal to' type. Therefore, all the points on the line are solutions to the inequality problem.

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Most popular questions from this chapter

Members of the group should interview a business executive who is in charge of deciding the product mix for a business. How are production policy decisions made? Are other methods used in conjunction with linear programming? What are these methods? What sort of academic background, particularly in mathematics, does this executive have? Present a group report addressing these questions, emphasizing the role of linear programming for the business.

Exercises 116-118 will help you prepare for the material covered in the next section. a. Graph the solution set of the system: $$\left\\{\begin{aligned} x+y & \geq 6 \\ x & \leq 8 \\ y & \geq 5 \end{aligned}\right.$$ b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(3 x+2 y\) at each of the points obtained in part (b).

Consider the objective function \(z-A x+B y \quad(A>0\) and \(B>0\) ) subject to the following constraints: \(2 x+3 y \leq 9, x-y \leq 2, x \geq 0,\) and \(y \geq 0 .\) Prove that the objective function will have the same maximum value at the vertices \((3,1)\) and \((0,3)\) if \(A-\frac{2}{3} B\).

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