/*! 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) Horizontal line at y = -2, (b) vertical line at x = 4, (c) x and y have opposite signs, (d) x or y is 0, (e) region above y = 1, (f) x-axis.

Step by step solution

01

Identify type of graph for equation y = -2

The equation \(y = -2\) represents a horizontal line in the coordinate plane. This line passes through all points where the y-coordinate is -2, regardless of the x-coordinate.
02

Identify type of graph for equation x = 4

The equation \(x = 4\) represents a vertical line in the coordinate plane. This line passes through all points where the x-coordinate is 4, regardless of the y-coordinate.
03

Analyze condition x/y < 0

The condition \(x / y < 0\) implies that the signs of x and y must be opposite for the inequality to hold. Thus, this condition describes the set of points where either x is positive and y is negative or x is negative and y is positive.
04

Analyze condition xy = 0

The condition \(xy = 0\) means that either x or y (or both) must be zero. This describes the set of points on the x-axis (y=0) and the y-axis (x=0), including the origin (0,0).
05

Identify region for inequality y > 1

The inequality \(y > 1\) describes the region above the horizontal line \(y = 1\) in the coordinate plane. This includes all points where the y-coordinate is greater than 1.
06

Identify type of graph for equation y = 0

The equation \(y = 0\) represents the x-axis itself, which includes all points where the y-coordinate is zero.

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 represented by an equation in the form of \(y = c\), where \(c\) is a constant. This means that no matter what value \(x\) takes, the \(y\)-value remains constant.

For example, in the exercise, the equation \(y = -2\) signifies that the line passes through the y-coordinate -2 across all values of \(x\). This creates a line that moves from left to right horizontally.

**Characteristics of Horizontal Lines:**
  • Parallel to the x-axis.
  • Do not cross or touch the y-axis, except at the specified y-value.
  • Every point on the line has the same y-value.
Horizontal lines are vital in understanding sections of inequalities or regions in a plane where the \(y\)-coordinate lies above, below, or exactly on a certain value.
Vertical Line
Vertical lines in the coordinate plane are defined by equations of the form \(x = c\), where \(c\) is a constant value for the x-coordinate, no matter what the y-coordinate is.

As in our exercise with \(x = 4\), this creates a line that intersects the x-axis at the point (4,0) and extends infinitely up and down. All the points on this line share the same x-coordinate.

**Characteristics of Vertical Lines:**
  • Parallel to the y-axis.
  • Every point on the line has the same x-value.
  • They do not have a slope (the slope is undefined).
Vertical lines can be crucial when defining boundaries in inequalities or when determining intersections with other lines in the coordinate plane.
Inequality
Inequalities describe a range or set of points that do not strictly lie on one particular line but satisfy a condition, like \(x/y < 0\). This means both signs must be opposite for the inequality to hold.

In simpler terms, the inequality states that:
  • If \(x\) is positive, \(y\) should be negative.
  • If \(x\) is negative, \(y\) should be positive.
The regions described by inequalities can be visualized in the coordinate plane as areas above, below, or in some specific relation to a line or boundary (such as \(y > 1\) indicating all \(y\)-values greater than 1).

Inequalities are foundational in linear programming, optimization tasks, and many real-world applications requiring range-based conditions.
Axes
Coordinate planes feature two primary reference lines: the x-axis and the y-axis.

**The x-axis** is the horizontal plane where \(y = 0\). Points on this axis have no vertical component. For example, \(y = 0\) in an equation defines the x-axis itself.

**The y-axis** is the vertical plane where \(x = 0\). Here, points have no horizontal component. Together, these axes intersect at the origin point, (0,0).

**Significance of Axes:**
  • They act as reference lines for other graphed lines or points.
  • Crossing points on these axes often determine specific lines or conditions, like \(xy = 0\) indicating points on either axis.
  • Used to divide the plane into quadrants aiding in graph analysis.
Understanding axes enables the interpretation of graphs and the positioning of lines or equations on the plane itself.

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 f. $$f(x)=\left\\{\begin{array}{ll} 3 & \text { if } x<-2 \\ -x+1 & \text { if }|x| \leq 2 \\ -4 & \text { if } x>2 \end{array}\right.$$

Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. Vertex \((4,-7), x\) -intercept \(-4\)

A 100 -foot-long cable of diameter 4 inches is submerged in seawater. Because of corrosion, the surface area of the cable decreases at the rate of 750 in \(^{2}\) per year. Express the diameter \(d\) of the cable as a function of time \(t\) (in years). (Disregard corrosion at the ends of the cable.)

Find a composite function form for \(y\) $$y=\frac{1}{(x-3)^{6}}$$

A spherical balloon is being inflated at a rate of \(\frac{9}{2} \pi \mathrm{ft}^{3} / \mathrm{min} .\) Express its radius \(r\) as a function of time \(t\) (in minutes), assuming that \(r=0\) when \(t=0\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure 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.