Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Name the possible quadrants in which the point ( \(x, y\) ) can lie if the given condition is true. (Hint: Consider the rules for determining the product and the quotient of signed numbers.) $$\frac{x}{y}<0$$

Step-by-Step Solution

Answer

Quadrants II and IV

Step 1: Understanding the Condition

The condition given is \(\frac{x}{y} < 0\). This means that the fraction is negative. We know a fraction is negative when one of the numbers is positive and the other is negative.

Step 2: Determining Signs of Coordinates

Since \(\frac{x}{y} < 0\), the signs of \(x\) and \(y\) must be opposite. That means either \(x > 0\) and \(y < 0\) or \(x < 0\) and \(y > 0\).

Step 3: Identifying Possible Quadrants

In the Cartesian coordinate system, when \(x > 0\) and \(y < 0\), points are in Quadrant IV. When \(x < 0\) and \(y > 0\), points are in Quadrant II.

Key Concepts

Signed Numbers

Signed numbers are numbers that include a positive or negative sign. They help us determine the position on various number lines, including those of the Cartesian plane. When handling signed numbers, these are some critical points to remember:

• Positive numbers are greater than zero and are usually written without a plus sign (e.g., 5, 12, and 89).
• Negative numbers are less than zero and always have a minus sign (e.g., -3, -17, and -58).

These concepts become crucial in understanding the negative fraction condition from the exercise. If the quotient \(\frac{x}{y} < 0\), it implies that one of the numbers x or y is positive, and the other is negative. This sign difference directly affects where the points \(x, y\) are plotted on the Cartesian plane.

Fraction Negativity

When a fraction like \(\frac{x}{y}\) is negative, it means that the numerator and denominator have opposite signs. Here's what happens in such cases:

• If the numerator is positive and the denominator is negative, the fraction becomes negative.
• Conversely, if the numerator is negative and the denominator is positive, the fraction remains negative.

Using this rule, we can determine the positional relationship of x and y on a coordinate plane. If \(\frac{x}{y} < 0\), it signifies that either \(x > 0\) and \(y < 0\) or \(x < 0\) and \(y > 0\). This fraction negativity is vital in understanding which quadrants these signed pairs will end up in, such as Quadrants II and IV in this specific instance.

Cartesian Coordinate System

The Cartesian coordinate system is an essential mathematical tool that allows us to pinpoint locations using a pair of numerical values. Here's how it works:

• The plane is divided into four quadrants by the x-axis (horizontal line) and y-axis (vertical line).
• Quadrant I (where \(x > 0\) and \(y > 0\)) is at the top-right, Quadrant II (where \(x < 0\) and \(y > 0\)) at the top-left, Quadrant III (where \(x < 0\) and \(y < 0\)) at the bottom-left, and Quadrant IV (where \(x > 0\) and \(y < 0\)) at the bottom-right.

With the condition \(\frac{x}{y} < 0\), knowing the signs of x and y guides us to place points correctly within these quadrants. If x is positive and y is negative, the point lands in Quadrant IV. Conversely, if x is negative and y is positive, the point finds itself in Quadrant II. Such spatial understanding parallels the rules for signed numbers and fraction negativity.