/*! 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 8 Describe the set of all points \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 8

Describe the set of all points \(P(x, y)\) in a coordinate plane that satisfy the given condition. (a) \(y=-2\) (b) \(x=-4\) (c) \(x / y<0\) (d) \(x y=0\) (e) \(y>1\) (f) \(y=0\)

Short Answer

Expert verified
(a) Points with \( y = -2 \); (b) Points with \( x = -4 \); (c) Points where \( x/y < 0 \); (d) Points on axes; (e) Points above \( y = 1 \); (f) Points on x-axis.

Step by step solution

01

Understanding Equation (a)

The equation given is \( y = -2 \). This equation represents a horizontal line on the coordinate plane where the y-value of every point is -2. Thus, the set of points satisfying this equation is all points \( P(x, y) \) where \( y = -2 \).
02

Understanding Equation (b)

The equation given is \( x = -4 \). This equation represents a vertical line on the coordinate plane where the x-value of every point is -4. Thus, the set of points satisfying this equation is all points \( P(x, y) \) where \( x = -4 \).
03

Understanding Inequality (c)

The inequality given is \( \frac{x}{y} < 0 \). This implies that the product \( xy < 0 \). Therefore, either \( x > 0 \) and \( y < 0 \) or \( x < 0 \) and \( y > 0 \). This describes the regions where the x and y coordinates have opposite signs.
04

Understanding Equation (d)

The equation given is \( xy = 0 \). This means that either \( x = 0 \) or \( y = 0 \). Thus, the set of all points satisfying this equation is the union of the x-axis (\( y = 0 \)) and the y-axis (\( x = 0 \)).
05

Understanding Inequality (e)

The inequality given is \( y > 1 \). This describes all points where the y-coordinate is greater than 1, which represents all points above the horizontal line \( y = 1 \) on the coordinate plane.
06

Understanding Equation (f)

The equation given is \( y = 0 \). This is the equation of the x-axis, meaning all points on this line have a y-coordinate of zero. Thus, the set of points is all points \( P(x, y) \) where \( y = 0 \).

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.

Horizontal Line
A horizontal line in the coordinate plane is characterized by the equation of the form \( y = c \), where \( c \) is a constant. This means that for every point on this line, the y-coordinate remains constant at the value \( c \), while the x-coordinate can be any real number.
For example, in the equation \( y = -2 \), the horizontal line crosses the y-axis at -2 and stretches infinitely in the direction of the x-axis.
  • This line shows a consistent height on the graph, and it's parallel to the x-axis.
  • Every point on this line has a coordinate of the form \( P(x, -2) \).
By understanding the properties of horizontal lines, you can easily identify and represent them on a coordinate plane.
Vertical Line
In contrast to a horizontal line, a vertical line has the equation \( x = c \), where \( c \) is a constant. For this type of line, the x-coordinate of every point on the line is fixed at \( c \), while the y-coordinate can vary.
For instance, in the equation \( x = -4 \), the vertical line passes through -4 on the x-axis and extends infinitely in the upward and downward direction.
  • This line is perpendicular to the x-axis and parallel to the y-axis.
  • All points on this line have the form \( P(-4, y) \).
Recognizing vertical lines helps you understand how certain equations reflect positions in the coordinate plane.
Inequality
Inequalities are a way to express regions on the coordinate plane where certain conditions hold instead of precise points. For example, the inequality \( \frac{x}{y} < 0 \) means that the product \( xy < 0 \). This indicates that the x and y values must have opposite signs either:
  • \(x > 0\) and \(y < 0\), which places the points in the fourth quadrant.
  • \(x < 0\) and \(y > 0\), which puts the points in the second quadrant.
The regions that satisfy this inequality are where x and y differ in sign, representing specific parts of the coordinate plane.
Understanding inequalities allows you to determine such areas, providing a broader view of how equations interact beyond straight lines.
Union of Axes
The "union of axes" concept occurs when an equation or condition effects both the x-axis and the y-axis simultaneously. The equation \( xy = 0 \) illustrates this by indicating that either \( x = 0 \) or \( y = 0 \). Their combined answer is known as the union of these two lines.
  • When \( x = 0 \), all points that lie on the y-axis are included.
  • When \( y = 0 \), all points on the x-axis are included.
This means that all points where the coordinates fall either on the x-axis or y-axis are part of the solution.
Understanding this concept helps in recognizing situations where solutions fall on two different planes coming together, thus including both axes in the coordinate plane.

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

A farmer wishes to put a fence around a rectangular field and then divide the field into three rectangular plots by placing two fences parallel to one of the sides. If the farmer can afford only 1000 yards of fencing, what dimensions will give the maximum rectangular area?

Algebraic methods were used to find solutions to each of the following equations. Now solve the equation graphically by assigning the expression on the left side to \(Y_{1}\) and the number on the right side to \(\mathbf{Y}_{2}\) and then finding the \(x\) -coordinates of all points of intersection of the two graphs. (a) \(x^{5 / 3}=32\) (b) \(x^{4 / 3}=16\) (c) \(x^{2 / 3}=-36\) (d) \(x^{144}=125\) (e) \(x^{1 / 2}=-27\)

The human camnonball In the 1940 s, the human cannonball stunt was performed regularly by Emmanuel Zacchini for The Ringling Brothers and Barnum \& Bailey Circus. The tip of the cannon rose 15 feet off the ground, and the total horizontal distance traveled was 175 feet. When the cannon is aimed at an angle of \(45^{\circ},\) an equation of the parabolic flight (see the figure) has the form \(y=a x^{2}+x+c\) (a) Use the given information to find an equation of the flight. (b) Find the maximum height attained by the human cannonball.

Cable TY fee A cable television firm presently serves 8000 households and charges \(\$ 50\) per month. A marketing survey indicates that each decrease of \(\$ 5\) in the monthly charge will result in 1000 new customers. Let \(R(x)\) denote the total monthly revenue when the monthly charge is \(x\) dollars. (a) Determine the revenue function \(R\) (b) Sketch the graph of \(R\) and find the value of \(x\) that results in maximum monthly revenue.

Quantity discount A company sells running shoes to dealers at a rate of \(\$ 40\) per pair if fewer than 50 pairs are ordered. If a dealer orders 50 or more pairs (up to 600 ), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum amount of money for the company?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.