Problem 13 - Algebra and Trigonometry

Question

Sketch the region given by the set. \(\\{(x, y) | x y<0\\}\)

Step-by-Step Solution

Answer

Shade Quadrants II and IV on the Cartesian plane.

Step 1: Understand the inequality

The inequality given is \(xy < 0\). This inequality means that the product of \(x\) and \(y\) must be negative. In terms of points on the Cartesian plane, it implies that either \(x\) or \(y\) must be negative, but not both.

Step 2: Determine which regions satisfy the inequality

For \(xy < 0\), there are two possible cases: (1) \(x > 0\) and \(y < 0\), or (2) \(x < 0\) and \(y > 0\). Each case creates a region in the Cartesian plane where the inequality holds.

Step 3: Identify specific quadrants on the plane

The Cartesian plane is divided into four quadrants: Quadrant I where \(x > 0, y > 0\); Quadrant II where \(x < 0, y > 0\); Quadrant III where \(x < 0, y < 0\); and Quadrant IV where \(x > 0, y < 0\). Given our conditions, the regions satisfying \(xy < 0\) are Quadrants II and IV.

Step 4: Sketch the region

On the Cartesian plane, shade Quadrant II, where \(x < 0\) and \(y > 0\), and Quadrant IV, where \(x > 0\) and \(y < 0\). These shaded areas represent the solution region for \(xy < 0\). The axes themselves do not belong to the solution set.

Key Concepts

Quadrants on the Cartesian Plane

The Cartesian plane is a two-dimensional plane defined by a horizontal axis (the x-axis) and a vertical axis (the y-axis). It is divided into four regions, called quadrants, by these two perpendicular lines. Each quadrant corresponds to a unique combination of positive or negative values for x and y.

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

Understanding which quadrant a point lies in helps us analyze equations and inequalities involving x and y. For example, knowing that a condition is only satisfied in specific quadrants can help visualize the solution on the plane.

Understanding Inequalities on a Plane

Inequalities are mathematical statements that relate the size or order of two objects. In a Cartesian plane, we often use inequalities to describe regions where particular conditions hold true. The inequality given in the exercise, formulated as \(xy < 0\), signifies that the product of x and y must be less than zero. This requires some interpretation:
- For \(xy < 0\), the product is negative when one variable is positive and the other is negative.
- This means that all points (x, y) where either x is positive and y is negative, or x is negative and y is positive, satisfy the inequality.Such inequalities help us identify specific regions on the plane where the conditions hold true, excluding others.

Product of Variables in the Plane

The product of any two numbers depends on their individual signs. When we talk about the product of variables \(x\) and \(y\) like in the inequality \(xy < 0\), we need to determine the sign of the result based on the signs of \(x\) and \(y\):

• If x and y have the same sign (both positive or both negative), their product is positive.
• If x and y have different signs (one is positive, the other is negative), their product is negative.

In the context of the Cartesian plane, this product rule is crucial. The combination of positive and negative values across each quadrant directly influences which regions satisfy specific inequalities like \(xy < 0\). This particular inequality is satisfied in the quadrants that have different signs for x and y, which means Quadrants II and IV.