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Chapter 7: Problem 91

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

Short Answer

Expert verified
In an inequality graph, a dashed line implies that the values on the line are not included in the solution set of the inequality. It's used for 'greater than' (>) or 'less than' (<) inequalities.

Step by step solution

01

Definition of Inequality

In the context of mathematics, an inequality is a relation between two expressions that may not be equal. It's often represented by symbols like '<', '>', '鈮�', '鈮�' etc.
02

Understanding the graphical representation of inequalities

When graphing inequalities, we use a dashed or solid line to distinguish between two scenarios. The line represents all the possible solutions to the inequality.
03

Deciphering the meaning of a dashed line

A dashed line is used when the values on the line are not included in the solution set of the inequality. Meaning, in the inequality 'y > f(x)' or 'y < f(x)', a dashed line indicates that the points actually on line y=f(x) are not solutions to the inequality.
04

Checking solutions for a dashed line inequality

To decide test points are solutions or not for an inequality represented by a dashed line, we choose the points not on the line, substitute them into the inequality. If the inequality holds true, then they are the solutions.

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a. \(A\) student earns \(\$ 15\) per hour for tutoring and \(\$ 10\) per hour as a teacher's aide. Let \(x=\) the number of hours each week spent tutoring and let \(y=\) the number of hours each week spent as a teacher's aide. Write the objective function that models total weekly earnings. b. The student is bound by the following constraints: \(\cdot\) To have enough time for studies, the student can work no more than 20 hours per week. \(\cdot\) The tutoring center requires that each tutor spend at least three hours per week tutoring. \(\cdot\) The tutoring center requires that each tutor spend no more than eight hours per week tutoring. 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 \(y\) are nonnegative. d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region. [The vertices should occur at \((3,0),(8,0),(3,17),\) and \((8,12) .]\) e. Complete the missing portions of this statement: The student can earn the maximum amount per week by tutoring for______ hours per week and working as a teacher's aide for______ hours per week. The maximum amount that the student can earn each week is $\$$ _____
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