/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Sketch the region that correspon... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 6

Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ 2 x-3 y \leq 7 $$

Short Answer

Expert verified
First, we convert the inequality into an equation: \(2x - 3y = 7\). Next, we solve for y: \(y = \frac{2}{3}x - \frac{7}{3}\). We use the origin (0,0) as a test point and determine that the shaded region should include the origin. After sketching the boundary line and shading the region, we find that the region is unbounded and has no corner points.

Step by step solution

01

Analyze the inequality and convert it to an equation

Initially, let's turn the inequality into an equation for easier plotting: \[ 2x - 3y = 7 \] Now our inequality is: \[ 2x - 3y \leq 7 \]
02

Determine the boundary line for the inequality

Now let's solve the equation, \(2x - 3y = 7\), for y: \[ y = \frac{2}{3}x - \frac{7}{3} \] This linear equation will help us draw the boundary line for the inequality region.
03

Determine the region represented by the inequality

To determine which side of the line (above or below) we should shade, we can use a test point. Since the inequality is \(\leq\), if the test point satisfies the inequality, we'll shade the same side as the test point. Conversely, if it doesn't, we'll shade the opposing side. Let's take the origin (0,0) as the test point. Plugging it into the inequality, we have: \[ 2(0) - 3(0) \leq 7 \] Since \(0 \leq 7\) is true, we'll shade the region that includes the origin.
04

Sketch the boundary line and region

Now we will sketch the boundary line (dashed because the inequality includes "equal to") using the equation: \[ y = \frac{2}{3}x - \frac{7}{3} \] We will also shade the region that includes the origin to represent the inequality.
05

Determine if the region is bounded or unbounded

Looking at our graph, we can see that the shaded region extends infinitely, which means the region is unbounded.
06

Identify any corner points

Since the region is unbounded, it doesn't have any corner points.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

We suggest the use of technology. Round all answers to two decimal places. Minimize \(c=5.45 x+y+1.5 z+w\) subject to \(\quad 5.12 x-y \quad+w \geq 1,000\) \(z+w \geq 2,000\) \(\quad 1.12 x+y \quad \leq 500\) \(\quad x \geq 0, y \geq 0, z \geq 0, w \geq 0 .\)

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and no vitamin \(C\). Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 45 percent of the U.S. Recommended Daily Allowance (RDA) of vitamin \(\mathrm{C}\) for infants. Each serving of Gerber Apple Banana Juice contains 60 calories and 120 percent of the U.S. RDA of vitamin \(\mathrm{C}\) for infants. \(^{42}\) The cereal costs \(10 \mathrm{~d} /\) serving, the dessert costs \(53 \mathrm{~d} /\) serving, and the juice costs 27 d/serving. If you want to provide your child with at least 120 calories and at least 120 percent of the U.S. RDA of vitamin \(\mathrm{C}\), how can you do so at the least cost? What are your shadow costs for calories and vitamin \(\mathrm{C}\) ?

Why is the simplex method useful? (After all, we do have the graphical method for solving LP problems.)

$$ \begin{array}{ll} \text { Maximize } & p=x+5 y \\ \text { subject to } & x+y \leq 6 \\ & -x+3 y \leq 4 \\ & x \geq 0, y \geq 0 . \end{array} $$

Your other friend Jason is telling everyone that if there is only one constraint in a standard linear programming problem, then you will have to pivot at most once to obtain an optimal solution. Is he correct? Explain.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.