Problem 19 - Algebra and Trigonometry

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: Understand the Inequalities

The region is defined by the inequalities \(|x| \leq 2\) and \(|y| \leq 3\). This means that the x-values in the region must be between -2 and 2 inclusive, and the y-values must be between -3 and 3 inclusive. It involves the absolute value function, which creates symmetries.

Step 2: Translate Absolute Value Inequalities

Convert the inequalities involving absolute values into standard inequalities. The inequality \(|x| \leq 2\) translates to \(-2 \leq x \leq 2\), and \(|y| \leq 3\) translates to \(-3 \leq y \leq 3\). This gives concrete boundaries for the x and y coordinates in the plane.

Step 3: Identify Boundary Lines

Identify the boundary lines for the region. For the x-axis, the boundaries are the vertical lines \(x = -2\) and \(x = 2\). For the y-axis, the boundaries are the horizontal lines \(y = -3\) and \(y = 3\). These lines form the perimeter of the region.

Step 4: Sketch the Region

Draw a coordinate plane. First, sketch the boundaries: vertical lines at \(x = -2\) and \(x = 2\), and horizontal lines at \(y = -3\) and \(y = 3\). The enclosed area is a rectangle, shaded to indicate the region that satisfies both inequalities.

Step 5: Verify the Region

Ensure that the region is correctly identified by checking a few points. Points like \((0, 0)\), \((2, 3)\), \((-2, -3)\), and \((1, 2)\) should lie within the rectangle, while points like \((3, 3)\) or \((-3, 0)\) should lie outside. This confirms the region is correctly understood as all points \((x, y)\) such that \(-2 \leq x \leq 2\) and \(-3 \leq y \leq 3\).

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is a fundamental concept in mathematics, providing a backdrop for graphing points, lines, and shapes. Imagine it as a large grid, resembling a piece of graph paper stretching infinitely in all directions.

It consists of two axes:

• The horizontal axis is known as the x-axis.
• The vertical axis is called the y-axis.

These axes intersect at a point called the origin, labeled as (0,0), which is the central reference point.

Every point in this grid system is represented by a pair of numbers, (x, y), where x denotes how far left or right from the origin, and y denotes how far up or down.
Think of the coordinate plane as a map helping you locate specific positions in space, making it possible to visualize and solve mathematical problems involving relationships between these coordinates.

The Role of Inequalities in Defining Regions

Inequalities are crucial in defining regions on the coordinate plane. They determine the boundaries and the inclusion or exclusion of certain areas.

In mathematics, inequalities express relationships where values are not exactly equal but fall within a defined range. For example, the inequalities \(|x| \leq 2\) and \(|y| \leq 3\) set specific limits for the variables involved.

• The inequality \(-2 \leq x \leq 2\) means the x-coordinates of any point are from -2 to 2 inclusive.
• The inequality \(-3 \leq y \leq 3\) means the y-coordinates range from -3 to 3 inclusive.

These ranges are essential, as they help sketch regions that meet specific criteria, forming areas that can appear as geometric shapes like rectangles, circles, or more complex forms on the graph.

Sketching Regions with Absolute Value Inequalities

When sketching regions that involve absolute value inequalities, the process begins by identifying the boundaries defined by these inequalities. Absolute values simplify inequalities by ensuring they cover symmetrical ranges around a central point.

Take the inequality \(|x| \leq 2\):

• This implies the region covers all x-values between -2 and 2.
• Similarly, \(|y| \leq 3\) indicates y-values stretch from -3 to 3.

To sketch, draw the vertical lines at x = -2 and x = 2 and the horizontal lines at y = -3 and y = 3.

These lines form a rectangle. All points within this shape satisfy both inequalities. Shading this area visually emphasizes the solution, clearly separating it from regions outside. By sketching such regions, we're able to visualize complex expressions easily, giving a tangible form to theoretical concepts and enhancing understanding.