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Chapter 0: Problem 15

Determine the quadrant(s) in which \((x, y)\) is located so that the condition(s) is (are) satisfied. \(x y>0\)

Short Answer

Expert verified
The condition \(x y > 0\) is satisfied in Quadrant I and Quadrant III.

Step by step solution

01

Analyze given condition

The given condition is \(x y>0\). This means that the product of x and y is positive.
02

Understand the Quadrant system

In Quadrant I, both x and y are positive, so their product would be positive. In Quadrant II, x is negative and y is positive, so their product would be negative. In Quadrant III, both x and y are negative, so their product would be positive since the multiplication of two negative numbers results in a positive number. In Quadrant IV, x is positive and y is negative, so their product would be negative.
03

Determine the Quadrants

From the analysis of the Quadrant system in step 2, the condition \(x y > 0\) will be met in Quadrant I where both x and y are positive and in Quadrant III where both x and y are negative.

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Key Concepts

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

Understanding the Quadrant System
The quadrant system is a way of dividing the coordinate plane into four distinct regions. Each region is called a quadrant, and they are labeled sequentially counterclockwise, starting from the top right corner.
Here is how you can understand each quadrant:
  • **Quadrant I**: This is the upper right part of the plane where both x and y coordinates are positive. This means any point in this area will have a positive x and y value.
  • **Quadrant II**: This is the upper left part of the plane. Here, the x coordinates are negative while the y coordinates are positive. Therefore, points in this quadrant have a negative x value and a positive y value.
  • **Quadrant III**: Located in the lower left corner, this quadrant has both x and y coordinates as negative. As such, any point will have both x and y negative here.
  • **Quadrant IV**: This is the lower right portion where x is positive and y is negative. Points in this region have a positive x value and a negative y value.
By understanding these characteristics, we can determine the properties of values like signs of products based on their quadrant location.
Exploring the Coordinate Plane
The coordinate plane is a two-dimensional space formed by two number lines intersecting perpendicularly at their zero points, known as the origin. These number lines are the x-axis (horizontal) and y-axis (vertical).
  • Both positive and negative numbers exist on the axes. Moving to the right on the x-axis increases positive x-values, while moving to the left decreases them into negative values.
  • Similarly, moving up on the y-axis increases positive y-values, while moving down decreases them into negative values.
  • The origin, which is at the center where the axes intersect, is denoted as (0, 0).
Understanding the coordinate plane is essential because it helps us identify the position of any point given in terms of its x and y values. This positioning is key to determining the quadrant and, consequently, the characteristics of the point.
Determining the Sign of the Product
When analyzing the product of two variables, x and y, we focus on whether their product is positive or negative. Let's break this down:
  • **Positive Product**: This occurs in two situations: when both numbers are positive or when both numbers are negative. In Quadrant I, both x and y are positive, making the product positive. In Quadrant III, both x and y are negative, but because multiplying two negatives results in a positive, the product remains positive.
  • **Negative Product**: A negative product happens when one number is positive and the other is negative. This situation arises in Quadrant II where x is negative and y is positive, and in Quadrant IV where x is positive and y is negative.
So, based on the product condition, we can deduce helpful information about the location of a point on the coordinate plane, specifically which quadrants are applicable. It helps predict the behavior and relationship of variables in mathematical analysis.

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Most popular questions from this chapter

In Exercises 33-38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$ g(x)=(x+5)^{3} $$

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