Problem 38 - A Graphical Approach to College Algebra

Question

Name the possible quadrants in which the point ( \(x, y\) ) can lie if the given condition is true. $$\frac{x}{y}>0$$

Step-by-Step Solution

Answer

The point \((x, y)\) can lie in Quadrant I or Quadrant III.

Step 1: Understanding the Condition

The condition given is \(\frac{x}{y} > 0\). This implies that the quotient of \(x\) divided by \(y\) is positive. Since the only way a fraction is positive is if both the numerator and denominator are positive, or both are negative, we will explore these scenarios.

Step 2: Scenario 1: Both x and y are positive

If both \(x\) and \(y\) are positive, then the point \((x, y)\) lies in Quadrant I. This is because, in the first quadrant, both coordinates are positive.

Step 3: Scenario 2: Both x and y are negative

If both \(x\) and \(y\) are negative, then the point \((x, y)\) lies in Quadrant III. This is because, in the third quadrant, both coordinates are negative.

Step 4: Conclusion of Solutions

Based on the above scenarios, the point \((x, y)\) can lie in either Quadrant I or Quadrant III for the condition \(\frac{x}{y} > 0\) to hold true.

Key Concepts

Quadrants

In coordinate geometry, the Cartesian plane is divided into four sections called quadrants. Each quadrant represents a unique combination of positive and negative values of the x and y coordinates. The quadrants are named in a counter-clockwise direction starting from the top-right corner, which is Quadrant I.

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, and y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, and y is negative.

Remember, the position of any point on this plane can be determined by the sign of its coordinates.

Positive and negative coordinates

Coordinates describe the position of a point on a plane and consist of an x (horizontal) value and a y (vertical) value. The signs of these values determine the quadrant in which the point lies.
When both x and y are positive, the point is in Quadrant I, as both coordinates are greater than zero.
Conversely, if both coordinates are negative, the point is located in Quadrant III, where numbers less than zero reflect both axes.

• Positive x and y → Quadrant I
• Negative x and y → Quadrant III

Understanding these signs is crucial in identifying the correct location of points on the Cartesian plane.

Algebraic conditions

Algebraic conditions can dictate the relationship between the x and y coordinates. For example, the condition \(\frac{x}{y} > 0\) suggests that the quotient of x over y is positive. This can only occur in two scenarios:

• Both the numerator \(x\) and the denominator \(y\) are positive, meaning the point lies in Quadrant I.
• Both \(x\) and \(y\) are negative, placing the point in Quadrant III.

This condition is a simple yet powerful tool to determine in which quadrants a point can lie based on its coordinates. By interpreting these conditions algebraically, it becomes easier to navigate and understand coordinate geometry.