Problem 57 - Precalculus with Limits

Question

In Exercises 53-60, write a system of inequalities to describe the region. Rectangle: vertices at \((4,3),(9,3),(9,9),(4,9)\)

Step-by-Step Solution

Answer

The system of inequalities describing the rectangular region defined by the points (4,3), (9,3), (9,9), and (4,9) is: \(4 \leq x \leq 9\) and \(3 \leq y \leq 9\).

Step 1: Identify boundaries

First, one needs to identify the minimum and maximum values for \(x\) and \(y\). Checking the points, it can be seen that the minimum \(x\) value is 4, the maximum \(x\) value is 9. Similarly, the minimum \(y\) value is 3, and the maximum \(y\) value is 9.

Step 2: Write the inequalities

The inequalities representing this rectangular region can now be written. Because the region includes the boundaries, the inequalities will be 'less than or equal to' or 'greater than or equal to'. For \(x\), the rectangle is bounded by 4 and 9, so the inequalities will be \(4 \leq x \leq 9\). For \(y\), the rectangle is bounded by 3 and 9, so the inequalities will be \(3 \leq y \leq 9\). Remember, these inequalities include their boundary points because they are 'less than or equal to' or 'greater than or equal to'.

Key Concepts

Rectangular Region

A rectangular region in coordinate geometry is defined by four vertices that form the corners of the rectangle. In this exercise, the vertices are given as

• (4,3)
• (9,3)
• (9,9)
• (4,9)

These points represent the corners of the rectangle, creating a region enclosed by straight lines. Each side of the rectangle runs parallel either to the x-axis or the y-axis.
This is a typical feature of standard rectangles in the coordinate plane. The sides are perfectly horizontal and vertical, making calculations straightforward.
Identifying the boundaries of this region will help write the corresponding system of inequalities.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a mathematical discipline that uses algebraic formulas to represent geometric concepts. Here, it is applied to determine the boundaries of a geometric figure such as a rectangle.
In our example, the points

• (4,3)
• (9,3)
• (9,9)
• (4,9)

allow us to see how the rectangle is placed on the coordinate plane. Each point is defined by an ordered pair \((x, y)\), indicating horizontal and vertical positions, respectively.
By knowing these coordinates, one can quickly deduce the range of x-values and y-values that define the rectangle's edges.

Inequality Representation

The inequality representation is used to describe the limits or borders of the rectangular region on a coordinate plane. The step-by-step process involves:

• Identifying the minimum and maximum values of both the x and y coordinates.
• Setting up inequalities to express these boundaries mathematically.

For the x-coordinates, the inequalities are:- \(4 \leq x \leq 9\)
This expresses that the horizontal limits of the rectangle run from x = 4 to x = 9.
Similarly, for the y-coordinates, the inequalities are:- \(3 \leq y \leq 9\)
These inequalities state that the vertical extent of the rectangle goes from y = 3 to y = 9.
Together, these inequalities form a system that fully defines the region enclosed by the rectangle.

Geometry Problems

Geometry problems involving rectangles often challenge one to correctly use inequalities to describe the bordered area. When solving such problems, it's crucial to:

• Identify all vertex points correctly.
• Pay attention to whether boundaries should be inclusive, which is indicated by "less than or equal" (≤) or "greater than or equal" (≥) signs.
• Ensure all aspects, such as heights and widths, fit the definition of a rectangle, meaning opposite sides are equal and parallel.

This systematic approach not only helps in accurately solving the problem but also allows for verification by checking that the right regions are enclosed.
By understanding these features, students can improve their skills in analyzing and solving fundamental geometry problems dealing with basic shapes like rectangles.