Problem 26 - Algebra and Trigonometry

Question

\(19-32\) Sketch the region given by the set. $$ \\{(x, y) | x y>0\\} $$

Step-by-Step Solution

Answer

The region includes Quadrant I and Quadrant III.

Step 1: Understand the Inequality

The exercise involves the set \(\{(x, y) | xy > 0\}\), which defines a region in the coordinate plane. The inequality \(xy > 0\) implies that the product of \(x\) and \(y\) must be positive.

Step 2: Determine the Quadrants

For the product \(xy > 0\) to hold, both numbers must be either positive or negative. This situation occurs in Quadrant I (where both \(x\) and \(y\) are positive) and in Quadrant III (where both \(x\) and \(y\) are negative).

Step 3: Visualize the Quadrants

Quadrant I is located in the upper-right of the Cartesian plane, while Quadrant III is in the lower-left. Plot these quadrants on the coordinate grid as the regions where the inequality holds true.

Step 4: Identify the Boundary Lines

The boundary lines are the lines \(x = 0\) and \(y = 0\), which form the axes. At these lines, the product \(xy\) is zero, which does not satisfy \(xy > 0\). Hence, these lines do not belong to the region.

Step 5: Sketch the Region

Draw the Cartesian plane and shade Quadrant I and Quadrant III. Clearly mark the boundary lines such that they are not included in the shaded areas to indicate that the points on the axes are not part of the region.

Key Concepts

Quadrants in Cartesian Plane

In the Cartesian plane, we divide the plane into four distinct areas called quadrants. Each of these quadrants is determined by the signs of the x and y coordinates. As a quick refresher:

• Quadrant I: Both x and y have positive values, located in the top-right quadrant of the plane.
• Quadrant II: x is negative and y is positive, found in the top-left.
• Quadrant III: Both x and y values are negative, placed in the bottom-left quadrant.
• Quadrant IV: x is positive and y is negative, situated in the bottom-right.

Every point on the plane falls into one of these quadrants, or on the boundary axes which are not part of any quadrant. For the problem at hand, the inequality requires us to look at Quadrants I and III, where the product of x and y is positive, either because both are positive or both are negative.

Inequalities and Regions

The use of inequalities like \(xy > 0\) helps define which regions of the Cartesian plane we should focus on. This inequality simply tells us that we want the area where x and y multiply to a value greater than zero.

• If both x and y are positive, their product is positive.
• Similarly, if both are negative, their product remains positive since a negative times a negative yields a positive result.

With this reasoning, we narrow down our area of interest to Quadrants I and III. The line boundaries, specifically the axes, are where \(xy = 0\) happens. Since we're looking for areas where \(xy > 0\), these lines are the boundaries but not included in the solution region.

Sketching Graphs

Once we've determined the regions based on the inequalities, the next step is to graphically represent them, a crucial step for visual understanding. Start by drawing the x and y axes on the Cartesian plane. These lines are where x or y equals zero, serving as our boundaries.
Your main focus is now to shade the regions in Quadrants I and III:

• Quadrant I: Marked in the upper-right, shade this as it contains positive x and y values.
• Quadrant III: Located in the lower-left, this will also be shaded because x and y are both negative here, making their product positive.
• Ensure that the boundary lines along the x and y axes remain uncolored because they represent points where the product is not greater than zero.

Shading these regions clearly shows where the conditions of the inequality are met, with non-shaded areas and axes left untouched to communicate that they fall outside the solution set.