Problem 32 - Precalculus Mathematics for Calculus

Question

Sketch the region given by the set. $$\\{(x, y)|| x | \leq 2 \text { and }|y| \leq 3\\}$$

Step-by-Step Solution

Answer

The region is a rectangle with vertices at (-2, -3), (-2, 3), (2, -3), and (2, 3).

Step 1: Understanding the Set Description

The set \( \{(x, y) \,|\, |x| \leq 2 \text{ and } |y| \leq 3 \} \) describes a region in the plane where both \(x\) and \(y\) have absolute values less than or equal to specific numbers. The inequality \(|x| \leq 2 \) means the x-values are between \(-2\) and \(2\). Similarly, \(|y| \leq 3\) means the y-values are between \(-3\) and \(3\).

Step 2: Identifying the Shape of the Region

Since \(|x| \leq 2\) and \(|y| \leq 3\), this creates a rectangle in the coordinate plane. The rectangle's boundaries are given by the equations: \(x = -2\), \(x = 2\), \(y = -3\), and \(y = 3\).

Step 3: Sketching the Region on the Coordinate Plane

Draw the coordinate plane with the x-axis ranging from -2 to 2 and the y-axis ranging from -3 to 3. The region is bounded by the vertical lines \(x = -2\) and \(x = 2\) as well as the horizontal lines \(y = -3\) and \(y = 3\). Shade the rectangle formed by connecting these boundary lines.

Key Concepts

Absolute Value

In mathematics, the absolute value of a number essentially represents its distance from zero on the number line, regardless of direction. It is denoted by vertical bars, like this: \(|x|\). If \(x\) is a positive number or zero, \(|x|\) is simply \(x\). If \(x\) is negative, \(|x|\) is its positive counterpart. For example:

• \(|5| = 5\)
• \(|-5| = 5\)

In the set given in the exercise, \(|x| \leq 2\) means that \(x\) can range from \(-2\) to \(2\). Similarly, \(|y| \leq 3\) means \(y\) can range from \(-3\) to \(3\). Understanding absolute values is crucial, as they define the bounds and interval sizes when sketching regions on the coordinate plane.

Inequalities

Inequalities are mathematical expressions that define the relative size or order of two values. In our exercise, we encountered the inequalities \(|x| \leq 2\) and \(|y| \leq 3\). These inequalities mean that \(x\) and \(y\) are limited to certain ranges on the coordinate plane.

• \(|x| \leq 2\) implies \(-2 \leq x \leq 2\).
• \(|y| \leq 3\) implies \(-3 \leq y \leq 3\).

In general, inequalities are used to define regions, establish limits, or specify conditions in mathematical problems.They play a vital role in forming a region on a coordinate plane, like a rectangle in this exercise.

Coordinate Plane

The coordinate plane is a two-dimensional plane where each point is uniquely specified by a pair of numerical coordinates. These coordinates are distances from two fixed perpendicular directed lines, typically called the x-axis (horizontal) and y-axis (vertical).In our problem, the coordinate plane helps us visualize the region defined by the set of inequalities. By understanding how to plot points, lines, and shapes on this grid, we can visually interpret the given data or conditions. When sketching regions like our exercise demands, ensure to:

• Mark boundary lines, like \(x = -2\), \(x = 2\), \(y = -3\), and \(y = 3\).
• Shade or highlight the complete region formed by these boundaries.

This visualization aids in grasping where all the solutions lie within this defined range on the plane.

Rectangular Region

A rectangular region on the coordinate plane is defined by its boundary lines. These lines set the limits for both the x and y-coordinates, creating a space where the constraints of a problem are satisfied. In our exercise, the region is described by \(|x| \leq 2\) and \(|y| \leq 3\), which translates to this specific rectangular area:

• Horizontally from \(x = -2\) to \(x = 2\).
• Vertically from \(y = -3\) to \(y = 3\).

The resulting rectangle has sides parallel to the x and y-axes. When sketching:

• Draw lines at each boundary.
• Shade the inside area to define the region.

Recognizing these rectangular formations can simplify understanding complex geometric constraints often found in coordinate geometry.