Problem 59 - Introductory Algebra for College Students

Question

Write a system of inequalities that describes all the points in quadrant III of a rectangular coordinate system.

Step-by-Step Solution

Answer

The system of inequalities that covers all the points in quadrant III of a rectangular coordinate system is given by \(x < 0\) and \(y < 0\).

Step 1: Understand Quadrant III

In a rectangular coordinate system or XY-plane, Quadrant III is the bottom left quadrant. Here, both x- and y-coordinates are negative.

Step 2: Formulate conditions for quadrant III

Based on the structure of the rectangular coordinate system, any point located in Quadrant III has negative values of x and y. Thus, for every point (x, y) in this quadrant, x < 0 and y < 0.

Step 3: Writing System of Inequalities

Having understood the conditions for a point to be in Quadrant III, the system of inequalities which represents all the points in Quadrant III is given as \(x < 0\) and \(y < 0\).

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is one of the most fundamental systems used in mathematics to represent geometric figures and positions. It consists of a two-dimensional plane defined by two axes: the horizontal x-axis and the vertical y-axis.
The point where these axes intersect is called the origin, denoted as (0, 0). Each point in the plane is represented by an ordered pair of numbers (x, y), which describe its location in relation to the x and y axes.
The plane is divided into four sections, known as quadrants:

• Quadrant I: where both x and y are positive.
• Quadrant II: where x is negative, and y is positive.
• Quadrant III: where both x and y are negative.
• Quadrant IV: where x is positive, and y is negative.

Understanding this system is crucial as it lays the groundwork for more advanced mathematical concepts and geometry problem-solving.

System of Inequalities

In mathematics, a system of inequalities consists of two or more inequalities that are solved simultaneously. They are often used to describe ranges or areas in a coordinate plane.
For instance, if you want to express a region of a plane where certain conditions are met, such as a particular quadrant in the rectangular coordinate system, you can do so using inequalities.

To describe Quadrant III, you use the inequalities:

• The inequality for the x-coordinate is given by: \(x < 0\).
• The inequality for the y-coordinate is: \(y < 0\).

These inequalities together form a system that describes all points with both x and y being negative. Understanding how to set up and solve systems of inequalities is essential for solving real-world problems, such as optimization and identifying feasible regions for solutions.

Negative Coordinates

Coordinates are values that represent a point's location on a plane, using ordered pairs (x, y). In terms of coordinates, negative numbers indicate a position in one of the quadrants where either or both values deviate to one axis's negative direction.
In Quadrant III of the rectangular coordinate system, both x and y coordinates are negative. This means every point in Quadrant III has an ordered pair such as (-3, -2) or (-5, -7).

Negative coordinates are crucial for understanding how points are distributed across different quadrants:

• They indicate the direction and distance from the origin but in a negative sense.
• The negative x-values show points that are to the left of the y-axis.
• Negative y-values indicate points that are below the x-axis.

Understanding negative coordinates helps in plotting points accurately and is key for graphing functions and interpreting geometric figures.