Problem 61 - Calculus Single Variable

Question

Sketch the set. \(\\{(x, y):|x| \cdot|y|<1\\}\)

Step-by-Step Solution

Answer

The set is the interior region bounded by the hyperbolas \(xy = 1\) and \(xy = -1\) in all quadrants.

Step 1: Understand the Inequality

The given inequality is \(|x| \cdot |y| < 1\). This inequality implies that the product of the absolute values of \(x\) and \(y\) must be less than 1.

Step 2: Consider Quadrants

Since both \(x\) and \(y\) can be positive or negative, we need to consider all four quadrants of the coordinate plane for this inequality.

Step 3: Visualize the Boundary

The boundary of the set \(|x| \cdot |y| = 1\) corresponds to the hyperbolas \(xy = 1\) and \(xy = -1\) in all four quadrants. Sketch these curves for reference.

Step 4: Shade the Solution Region

The region satisfying \(|x| \cdot |y| < 1\) is inside the region bounded by the hyperbolas \(|x| = \frac{1}{|y|}\) and \(|y| = \frac{1}{|x|}\). Shade the inside of the hyperbolas in each quadrant.

Step 5: Identify Excluded Areas

The curves \(xy = 1\) and \(xy = -1\) are not included in the solution set, so the lines themselves should remain unshaded.

Key Concepts

Coordinate Plane

In mathematics, the coordinate plane is a fundamental concept used in geometry and calculus. It consists of two number lines that intersect at a right angle: one horizontal (the x-axis) and one vertical (the y-axis). The point where these axes intersect is known as the origin, denoted by the coordinates (0, 0). The purpose of the coordinate plane is to provide a framework for plotting ordered pairs of numbers.

• The first number in an ordered pair corresponds to the position on the x-axis.
• The second number corresponds to the position on the y-axis.

By using these coordinate axes, we can divide the plane into four sections, called quadrants. These quadrants are labeled counterclockwise starting from the upper right:

• 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.

Mapping points and graphing equations on this plane helps us visualize mathematical relationships. It's especially useful when solving inequalities or sketching graphs like hyperbolas.

Inequality

Inequalities express the relationship between two values, indicating that one value is larger or smaller than another. In mathematical notation, inequality symbols like <, >, ≤, and ≥ are used. In the given exercise, we are dealing with an inequality that involves absolute values: \(|x| \cdot |y| < 1\).This tells us that the product of the absolute values of x and y must be less than 1. Absolute values are always non-negative, which means

• We're interested in all x and y values such that their absolute product is less than 1.
• This product is symmetric across all four quadrants of the coordinate plane.

To solve inequalities like this, it's essential to grasp the concept of absolute value and how operations like multiplication affect them. Consideration of different regions or sections of the coordinate plane often aids in solving such inequalities and determining the solution set.

Hyperbola

A hyperbola is a type of conic section defined as the set of points whose difference in distances from two fixed points (foci) remains constant. In the exercise, the inequality boundary \(|x| \cdot |y| = 1\) corresponds to hyperbolas, specifically given by the equations:

• \(xy = 1\)
• \(xy = -1\)

These hyperbolas appear in all four quadrants of the coordinate plane. Each hyperbola comprises two branches, opening away from the origin. They mirror one another across asymptotes, which are the axes and lines that pass through the origin in the case of these equations. In our problem, the lines we sketch (\( xy = 1\) and \( xy = -1\)) show the boundary conditions of the regions where we have \(|x| \.\ |y| < 1\).Visualizing these hyperbolas helps us to distinguish regions where the solution inequality is true. This involves shading the interior of these hyperbola arcs, thereby excluding the boundaries, which are not part of our solution set.