Problem 23 - Precalculus with Limits

Question

In Exercise 15-24, determine the quadrant(s) in which \( (x, y) \) is located so that the condition(s) is (are) satisfied. \( xy > 0 \)

Step-by-Step Solution

Answer

The condition xy > 0 is satisfied in the first and the third quadrant.

Step 1: Analyze the given condition

Analyze the condition xy > 0, which states that the product of the coordinates x and y must be greater than 0.

Step 2: Apply the condition to each quadrant

Apply the given condition to each quadrant in the Cartesian Coordinate System. In the first quadrant, since both x and y are positive, xy > 0 is satisfied. In the second quadrant, x is negative whereas y is positive, which does not satisfy xy > 0. In the third quadrant, both x and y are negative, so the product is positive and xy > 0 is satisfied. In the fourth quadrant, x is positive and y is negative which does not satisfy xy > 0.

Step 3: Determine the quadrants

From step 2 we can observe that the condition xy > 0 is satisfied only in the first and the third quadrant.

Key Concepts

Quadrants

In the Cartesian Coordinate System, the coordinate plane is divided into four distinct regions known as quadrants. These are defined based on the positive and negative values of the x and y coordinates. Each quadrant is created by the intersection of the x-axis and y-axis at the origin, which is the point (0,0).

• First Quadrant: Both x and y values are positive (x > 0, y > 0).
• Second Quadrant: x is negative while y is positive (x < 0, y > 0).
• Third Quadrant: Both x and y values are negative (x < 0, y < 0).
• Fourth Quadrant: x is positive while y is negative (x > 0, y < 0).

Understanding these quadrants is crucial when analyzing conditions like inequalities on the coordinate plane. Each quadrant offers a different set of values that can affect the outcome of specific conditions, such as when analyzing which areas satisfy particular inequalities. This division helps in systematically identifying locations and solving equations based on x and y values.

Coordinate Plane

The coordinate plane is a two-dimensional surface formed by the intersection of a horizontal line, called the x-axis, and a vertical line, called the y-axis. This plane is fundamental in graphing mathematical equations and analyzing their solutions.

Each point on the coordinate plane is represented by an ordered pair (x, y), indicating its position relative to the x and y axes. The x-coordinate shows how far left or right the point is from the origin, and the y-coordinate indicates its vertical distance.

• Positive x-values are to the right of the origin, negative to the left.
• Positive y-values are above the origin, negative below.

The coordinate plane is divided into four quadrants, helping to pinpoint locations and analyze situations why any x and y value product—like when examining inequalities such as xy > 0. Understanding the coordinate plane is foundational for working with critical mathematical concepts.

Inequalities

In mathematics, inequalities are statements that express the relative size or order of two values. They are used to determine whether a number is greater or smaller than another. The concept is fundamental for problem-solving in the coordinate system, particularly in identifying points that satisfy specific conditions.

The expression xy > 0 describes a condition where the product of x and y is positive. To explore this:

• If both x and y are positive, their product is positive, satisfying the condition (found in the first quadrant).
• If both x and y are negative, their product is also positive, thus satisfying the condition (found in the third quadrant).
• Conversely, if x and y have opposite signs, such as one being positive and the other negative, xy becomes negative, which doesn't satisfy xy > 0.

By understanding how inequalities work with coordinates, students can effectively determine which quadrants fulfill particular conditions set by equations or expressions like the given inequality.