Problem 35 - 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.) $$x y>0$$

Step-by-Step Solution

Answer

The point \((x, y)\) can lie in Quadrants I and III.

Step 1: Understanding the Condition

We need to determine in which quadrants the point \((x, y)\) can lie given that \(xy > 0\). This condition means that the product of \(x\) and \(y\) is a positive number.

Step 2: Analyzing Positive Product

The product \(xy > 0\) implies that both \(x\) and \(y\) have the same sign. So, either both \(x\) and \(y\) are positive, or both \(x\) and \(y\) are negative.

Step 3: Identifying Quadrants with Positive Coordinates

In the coordinate plane, both \(x\) and \(y\) are positive in the first quadrant (Quadrant I), where \((x,y) > 0\).

Step 4: Identifying Quadrants with Negative Coordinates

In the coordinate plane, both \(x\) and \(y\) are negative in the third quadrant (Quadrant III), where \((x,y) < 0\).

Step 5: Conclusion on Possible Quadrants

Since the point can have both positive or both negative coordinates to satisfy \(xy > 0\), the point \((x, y)\) can lie in Quadrants I and III.

Key Concepts

Understanding Quadrants in the Coordinate Plane

The coordinate plane is divided into four sections, or quadrants, by the x-axis and y-axis. These quadrants help us determine the location of any point with coordinates \(x, y\). Each quadrant is defined by the signs of the x and y coordinates:

• Quadrant I: Both coordinates are positive (\(x > 0, y > 0\)).
• Quadrant II: x is negative and y is positive (\(x < 0, y > 0\)).
• Quadrant III: Both coordinates are negative (\(x < 0, y < 0\)).
• Quadrant IV: x is positive and y is negative (\(x > 0, y < 0\)).

When analyzing which quadrant a point falls into, it's essential to check the signs of \(x\) and \(y\). Each combination leads to a unique quadrant location.

The Role of Signed Numbers

Signed numbers are numbers that have either a positive or negative sign. This sign indicates the number’s direction relative to zero on a number line.

• A positive number is greater than zero.
• A negative number is less than zero.

In math, the product or quotient of signed numbers behaves according to specific rules:

• Product of two positive numbers is positive.
• Product of two negative numbers is positive.
• Product of a positive and negative number is negative.
• Quotient follows the same rules as the product.

Understanding these rules helps in determining conditions like \(xy > 0\), where the signs of x and y must match.

Positive and Negative Coordinates

Coordinates are a pair of numbers expressed as \(x, y\) that describe a point's position on the coordinate plane. The sign of these coordinates determines which quadrant they reside in.

• If both coordinates are positive, the point is in Quadrant I.
• If both coordinates are negative, the point is in Quadrant III.

The condition \(xy > 0\) means that the product of x and y is positive. For this to happen:

• Both x and y must be positive, placing the point in Quadrant I.
• Both x and y must be negative, placing the point in Quadrant III.

This aligns with the behavior of signed numbers and confirms which quadrants the point can lie in. The mathematical analysis involves confirming that the quadrant aligns with the signs of the coordinates, ensuring accurate placement on the plane.