Problem 24 - 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 point (x, y) would be located in either the second or fourth quadrant.

Step 1: Understanding the properties of quadrants

In the Cartesian coordinate system, every point is represented as (x, y). In the first quadrant, both x and y are positive. In the second quadrant, x is negative and y is positive. In the third quadrant, both x and y are negative. While in the fourth quadrant, x is positive and y is negative.

Step 2: Applying the given condition

The exercise provides the condition \( xy < 0 \). Let's analyze this: In the first quadrant (both x and y are positive), the product \( xy \) would be positive which doesn't satisfy the condition. In the second quadrant (x is negative and y is positive), the product \( xy \) would be negative which satisfies the condition. In the third quadrant (both x and y are negative), the product \( xy \) would again be positive which doesn't satisfy the condition. Lastly, in the fourth quadrant (x is positive while y is negative), the product \( xy \) would be negative, which satisfies the condition.

Step 3: Conclusion on the position of (x, y)

From the above analysis, we can conclude that the condition \( xy < 0 \) is satisfied if the point (x, y) lies in the second or fourth quadrant.

Key Concepts

Quadrants of Coordinate Plane

The Cartesian coordinate system is a fundamental framework for representing points in the plane using two perpendicular lines called axes, which intersect at a point called the origin. This system divides the plane into four regions known as quadrants. Understanding quadrants is essential in geometry, algebra, and even calculus.

Each quadrant is denoted by a Roman numeral and is unique in the signs of the coordinates it contains:

• In Quadrant I (or the first quadrant), both the x (horizontal) and y (vertical) values are positive.
• Quadrant II (or the second quadrant) has negative x values and positive y values.
• Quadrant III (or the third quadrant) contains points with both negative x and negative y values.
• In Quadrant IV (or the fourth quadrant), the x values are positive whereas y values are negative.

Knowing which quadrant a point lies in can help solve various mathematical problems, such as determining the sign of an expression involving coordinates, like the product of the x and y values.

Signs of Coordinates

The signs of a point's coordinates tell us a lot about where a point is located in the Cartesian plane. Depending on the quadrant, the signs of the x and y coordinates can be either positive or negative:

First Quadrant

Both x and y are greater than zero, hence both are positive.

Second Quadrant

Here, x is less than zero (negative), while y is greater than zero (positive).

Third Quadrant

In this quadrant, both x and y are less than zero, making them both negative.

Fourth Quadrant

Finally, x is greater than zero (positive) and y is less than zero (negative).

Points on the axes themselves are a special case where either x or y (but not both) can be zero. These details help us solve problems involving expressions that include coordinates, as the signs dictate whether these expressions will be positive or negative in value.

Inequalities in Coordinate Plane

Inequalities such as the one from the exercise, where we have an inequality involving the multiplication of x and y (mathbf{xy < 0}), are very informative. An inequality that states 'mathbf{xy < 0}' implies that the signs of x and y coordinates must be opposite to each other because only the multiplication of one positive and one negative number result in a negative product.

This information aligns with what we know about the quadrants:

• In the second quadrant, as x is negative and y is positive, their product is negative, satisfying the 'mathbf{xy < 0}' inequality.
• In the fourth quadrant, the positive x and negative y also result in a negative product, which again satisfies the inequality.

At no point in the first or third quadrants would their multiplication satisfy 'mathbf{xy < 0}.' These insights are highly beneficial when solving more complex problems including inequalities with coordinate points.