/*! 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 38 Graph the solution set of the sy... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 38

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x<2 y-y^{2} \\ 0

Short Answer

Expert verified
The solution to the system of inequalities is the region on the left side of the y-axis (x<0), below the curve of the parabola and above the line.

Step by step solution

01

Graph the First Inequality \(x

To graph this inequality, it might be easier to rewrite it in the form of \(x=y(2-y)\) as \(y=\sqrt{2-x}\) and \(y=-\sqrt{2-x}\). It's a downwards facing parabola once graphed. However, keep in mind that the inequality is 'less than', so all the x-values under the curve of the parabola are included to the solution set of this inequality.
02

Graph the Second Inequality \(0

To graph the second inequality, it would be easier if we rewrite it as \(y<-x\). It'll be a straight line passing through origin, with a negative slope. However, since the inequality is 'greater than', the solution set for this inequality will be the area above the line.
03

Find the Overlapping Region

Now that the graphs of both inequalities have been plotted on the Cartesian plane, the common solution would be the overlapping region or intersection of the two solution sets. You can clearly see the overlapping region on the left side of the y-axis (i.e. when x<0).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Inequalities
When it comes to understanding systems of inequalities, graphing is a powerful visual tool. It helps in observing not just individual solutions, but also the relationships between multiple inequalities. Each inequality in a system is graphed separately on a coordinate plane. The combined solution is found in the intersection of their solution sets.
  • Start by transforming inequalities into equations to sketch their graphs.
  • Use solid or dashed lines depending on whether the inequality sign includes equal to (鈮� or 鈮� for solid, < or > for dashed lines).
  • For each inequality, shade the area representing solutions; it indicates which side of the line or curve satisfies the inequality.
  • The overlapping shaded region of all inequalities represents the final solution set.
This visual approach not only enhances understanding but also fosters intuition about how inequalities interact and the nature of their solutions.
Parabola
A parabola is a U-shaped curve that is the graph of a quadratic function or equation. It has some distinctive characteristics:
  • It can open upwards or downwards depending on the sign of the leading coefficient.
  • The highest or lowest point of a parabola is known as the vertex.
  • The axis of symmetry passes through the vertex and divides the parabola into two mirror images.
  • A downward-opening parabola is typically represented by equations like \(y = ax^2 + bx + c \) where \(a < 0\).
  • For the inequality \(x < 2y - y^2\), rewriting it as \(x = y(2-y)\) helps to visualize it as a down-facing parabola, indicating that solutions lie beneath the curve.
Understanding the shape and properties of a parabola makes it easier to approach graphing tasks, particularly when determining solution regions.
Linear Inequality
Linear inequalities involve expressions like ax + by < c, similar to linear equations but instead use inequality signs (<, >, 鈮�, 鈮�). Graphing linear inequalities involves a few steps that make it clear where solutions lie:
  • Begin by graphing the related equation (e.g., y = mx + b) which serves as the boundary line.
  • Convert the inequality into a form that makes graphing intuitive; for example, from \(0 < x+y\) to \(y < -x\), indicating the line crosses through the origin with a negative slope.
  • An inequality like \(y < -x\) means shading below the line, as this area contains all the points where x and y satisfy the inequality.
  • Since this line divides the plane, the solution region is either the half-plane above or below the boundary line.
Recognizing how linear inequalities shape solution sets enhances problem-solving abilities and makes it simple to interpret or construct graphs of systems of inequalities.

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

Sketch the graph of the inequality. $$2 y-x \geq 4$$

Find the consumer surplus and producer surplus for the pair of demand and supply equations. Supply \(p=125+0.0006 x\) Demand $$p=600-0.0002 x$$

Optimal Profit A manufacturer produces two models of bicycles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table. $$ \begin{array}{|l|c|c|} \hline \text { Process } & \text { Model A } & \text { Model B } \\ \hline \text { Assembling } & 2 & 2.5 \\ \hline \text { Painting } & 4 & 1 \\ \hline \text { Packaging } & 1 & 0.75 \\ \hline \end{array} $$ The total times available for assembling, painting, and packaging are 4000 hours, 4800 hours, and 1500 hours, respectively. The profits per unit are \(\$ 50\) for model \(\mathrm{A}\) and \(\$ 75\) for model \(\mathrm{B}\). What is the optimal production level for each model? What is the optimal profit?

Kayak Inventory A store sells two models of kayaks. Because of the demand, it is necessary to stock at least twice as many units of model \(\mathrm{A}\) as units of model \(\mathrm{B}\). The costs to the store for the two models are \(\$ 500\) and \(\$ 700\), respectively. The management does not want more than \(\$ 30,000\) in kayak inventory at any one time, and it wants at least six model A kayaks and three model B kayaks in inventory at all times. (a) Find a system of inequalities describing all possible inventory levels, and (b) sketch the graph of the system.

You have a total of $$\$ 500,000$$ that is to be invested in (1) certificates of deposit, (2) municipal bonds, (3) blue-chip stocks, and (4) growth or speculative stocks. How much should be put in each type of investment? The certificates of deposit pay \(2.5 \%\) simple annual interest, and the municipal bonds pay \(10 \%\) simple annual interest. Over a five-year period, you expect the blue-chip stocks to return \(12 \%\) simple annual interest and the growth stocks to return \(18 \%\) simple annual interest. You want a combined annual return of \(10 \%\) and you also want to have only one-fourth of the portfolio invested in stocks.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.