/*! 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 2 Sketch the region that correspon... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 2

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). $$ 4 x-y \leq 12 $$

Short Answer

Expert verified
The region corresponding to the inequality \(4x-y \leq 12\) is unbounded and has only one boundary line with the slope-intercept equation \(y=4x-12\). There are no corner points in the region.

Step by step solution

01

Rearrange the equation to slope-intercept form

To rearrange the equation, we first isolate y: \(y=4x-12\) Now, we can see that the slope of our boundary line, m, is 4 and the y-intercept, b, is -12. #Step 2: Determine which side of the line contains the region that satisfies the inequality# To figure out which side of the line contains the region that satisfies the inequality, we can choose a test point (not on the line) and check whether the inequality holds true for that point. If it does, then the region that contains this test point satisfies the inequality.
02

Choose a test point and plug it into the inequality

A common test point to use is the origin (0,0) if it does not lie on the boundary line. In this case, the origin is not on the line, so we can use it as our test point. Plug the origin into the inequality: \(4(0) - (0) \leq 12\) \(0 \leq 12\) Since the inequality holds true for the test point (0,0), the region that contains this point is the region that satisfies the inequality. #Step 3: Sketch the region on the coordinate plane# Now we need to sketch the boundary line and shade the region that satisfies the inequality, based on our test point.
03

Sketch the line and shade the region

First, plot the y-intercept (-12) on the y-axis. Then, use the slope of 4 to plot additional points on the line. Since the slope is positive, go up 4 units and to the right 1 unit for each additional point. Now, draw the line connecting the plotted points. Since the inequality is less than or equal to (鈮�), it means that the line also belongs to the solution, and we will use a solid line to indicate this. Finally, shade the region that contains the origin to show the region that satisfies the inequality. #Step 4: Analyze the sketch to determine if the region is bounded or unbounded, and to find the coordinates of any corner points# By examining our sketch, we can make our conclusions about the region and corner points.
04

Determine if the region is bounded or unbounded

Since the shaded region continues indefinitely to the right, the region is unbounded.
05

Find the coordinates of all corner points

As there is only one boundary line, there are no corner points in our region. In conclusion, we have sketched the region corresponding to the inequality \(4x-y \leq 12\), determined that it is unbounded, and found that there are no 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

\(\nabla\) Transportation Scheduling Your publishing company is about to start a promotional blitz for its new book, Physics for the Liberal Arts. You have 20 salespeople stationed in Chicago and 10 in Denver. You would like to fly at least 10 into Los Angeles and at least 15 into New York. A round-trip plane flight from Chicago to LA costs \(\$ 195 ;{ }^{.37}\) from Chicago to \(\mathrm{NY}\) costs \(\$ 182 ;\) from Denver to LA costs \(\$ 395\); and from Denver to NY costs \(\$ 166\). How many salespeople should you fly from each of Chicago and Denver to each of LA and NY to spend the least amount on plane flights?

Can the value of the objective function remain unchanged in passing from one tableau to the next? Explain.

$$ \begin{aligned} \text { Minimize } & c=6 s+6 t \\ \text { subject to } & s+2 t \geq 20 \\ & 2 s+t \geq 20 \\ & s \geq 0, t \geq 0 . \end{aligned} $$

Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Maximize and minimize \(\quad p=x+2 y\) subject to \(\begin{aligned} & x+y \geq 2 \\ & x+y \leq 10 \\ & x-y \leq 2 \\\ & x-y \geq-2 . \end{aligned}\)

$$ \begin{array}{|r|c|c|c|c|c|} \hline & \begin{array}{c} \text { Creatine } \\ (g) \end{array} & \begin{array}{c} \text { Carbohydrates } \\ (g) \end{array} & \begin{array}{c} \text { Taurine } \\ (g) \end{array} & \begin{array}{c} \text { Alpha Lipoic } \\ \text { Acid }(\mathrm{mg}) \end{array} & \begin{array}{c} \text { Cost } \\ (\$) \end{array} \\ \hline \begin{array}{r} \text { Cell-Tech }^{\infty} \\ \text { (MuscleTech) } \end{array} & 10 & 75 & 2 & 200 & 2.20 \\ \hline \begin{array}{r} \text { RiboForce } \\ \mathbf{H P}^{*} \text { (EAS) } \end{array} & 5 & 15 & 1 & 0 & 1.60 \\ \hline \begin{array}{r} \text { Creatine } \\ \text { Transport }^{*} \\ \text { (Kaizen) } \end{array} & 5 & 35 & 1 & 100 & 0.60 \\ \hline \begin{array}{r} \text { Pre-Load } \\ \text { Creatine } \\ \text { (Optimum) } \end{array} & 6 & 35 & 1 & 25 & 0.50 \\ \hline \end{array} $$ (Compare Exercise 30 in Section 4.2.) You are thinking of combining RiboForce HP, Creatine Transport, and Pre-Load Creatine to obtain a 10 -day supply that provides at least 80 grams of creatine and at least 10 grams of taurine, but no more than 600 grams of carbohydrates and 2,000 milligrams of alpha lipoic acid. How many servings of each supplement should you combine to meet your specifications for the least cost?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

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.