Problem 37 - 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 Quadrants II and IV.

Step 1: Understanding the Condition

The inequality \(\frac{x}{y} < 0\) implies that the quotient of \(x\) over \(y\) is negative. This will occur when one of them is positive and the other is negative.

Step 2: Analyzing Sign Variation

In a coordinate plane, \(x > 0, y < 0\) results in a negative value, and \(x < 0, y > 0\) also results in a negative value. This aligns with the condition \(\frac{x}{y} < 0\).

Step 3: Identifying Quadrants

In the coordinate plane, the combinations \((x > 0, y < 0)\) place the point in the Fourth Quadrant, and \((x < 0, y > 0)\) place the point in the Second Quadrant.

Step 4: Conclusion

Therefore, the condition \(\frac{x}{y} < 0\) can be satisfied in Quadrant II (where \(x < 0, y > 0\)) and Quadrant IV (where \(x > 0, y < 0\)).

Key Concepts

Quadrants

In the coordinate plane, you will encounter four sections called quadrants. These quadrants are numbered counterclockwise, starting from the top right.

• Quadrant I: Here, both coordinates, \(x\) and \(y\), are positive \((x > 0, y > 0)\).
• Quadrant II: In this quadrant, \(x\) is negative and \(y\) is positive \((x < 0, y > 0)\).
• Quadrant III: Here, both \(x\) and \(y\) are negative \((x < 0, y < 0)\).
• Quadrant IV: In this section, \(x\) is positive and \(y\) is negative \((x > 0, y < 0)\).

Understanding which quadrant a point lies in helps visualize the relationships between the coordinates, especially when dealing with inequalities such as \(\frac{x}{y} < 0\).

Inequality

An inequality expresses a relationship between two values, showing that one value is smaller or greater than the other. In our context, the inequality \(\frac{x}{y} < 0\) indicates that the fraction of \(x\) divided by \(y\) is negative.

For the fraction to be negative, different sign combinations emerge:

• \(x > 0\) and \(y < 0\), because dividing a positive number by a negative one yields a negative result.
• \(x < 0\) and \(y > 0\), since dividing a negative number by a positive one also gives a negative result.

Inequalities like these help to understand how signs of numbers influence the outcome of operations involving division.

Sign Variation

Sign variation in a coordinate system explains how changes in signs of \(x\) and \(y\) affect their product or quotient. It's crucial to determine the quadrant of a point based on the signs of its coordinates.

When analyzing the inequality \(\frac{x}{y} < 0\), the sign of the quotient determines the potential quadrants:

• In Quadrant II, \(x\) is negative and \(y\) is positive \((x<0, y>0)\), fulfilling the condition of a negative quotient.
• Conversely, in Quadrant IV, \(x\) is positive and \(y\) is negative \((x>0, y<0)\), also resulting in a negative quotient.

Recognizing these patterns helps to quickly identify possible placements for any point based on its sign characteristics.