Problem 15 - Precalculus with Limits

Question

In Exercise 15-24, determine the quadrant(s) in which \( (x, y) \) is located so that the condition(s) is (are) satisfied. \( x > 0 \) and \( y < 0 \)

Step-by-Step Solution

Answer

The coordinates \( (x, y) \), with the conditions \( x > 0 \) and \( y < 0 \) are located in Quadrant IV

Step 1: Understand the Cartesian Coordinate System

In the Cartesian coordinate system, space is divided into four quadrants by two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical). Quadrant I is where both x and y are positive. Quadrant II is where x is negative and y is positive. Quadrant III is where both x and y are negative. Quadrant IV is where x is positive and y is negative.

Step 2: Apply the Given Conditions

The problem stipulates that \( x > 0 \) and \( y < 0 \). This means that x is positive and y is negative. Considering the way the quadrants have been defined, one can immediately tell this falls into Quadrant IV.

Key Concepts

Understanding Quadrants in the Cartesian Coordinate System

The Cartesian Coordinate System is a mathematical framework that allows us to visualize equations and inequalities in a two-dimensional space. In this system, the plane is divided into four sections, known as quadrants, by the x-axis and y-axis. Each quadrant is a unique combination of positive and negative values for x and y.

• Quadrant I: Here, both x and y coordinates are positive. This is the upper-right section of the coordinate plane.
• Quadrant II: In this quadrant, x is negative, while y is positive, placing it in the upper-left part.
• Quadrant III: Both x and y are negative, placing this quadrant in the lower-left section of the plane.
• Quadrant IV: X values are positive, while y values are negative, situating this quadrant in the lower-right section.

When solving problems involving the location of points, understanding which quadrant conditions place you in is crucial. With coordinates given, such as \(x > 0\) and \(y < 0\), we pinpoint the quadrant as Quadrant IV, where these conditions are satisfied.

Navigating Coordinate Planes

A coordinate plane is a flat, two-dimensional surface used in mathematics for graphing equations and inequalities. It consists of two axes: the horizontal x-axis and the vertical y-axis. These axes intersect at the origin, a central point represented by \((0, 0)\). The intersection divides the plane into four quadrants.

Each point on the coordinate plane is identified by an ordered pair \((x, y)\). This pair tells us how far to move along the x-axis and y-axis from the origin. Understanding this system allows us to graphically represent algebraic equations and analyze spatial relationships between geometric shapes.
Furthermore, the coordinate plane is essential in identifying the location of points that satisfy specific conditions. By identifying the quadrant, mathematicians can determine where points meeting certain conditions are plotted.

Interpreting Inequalities in Coordinate Systems

Inequalities in the Cartesian Coordinate System help us determine regions on the plane where certain conditions are fulfilled. Unlike equations that define a specific line or curve, inequalities describe areas that include all points satisfying the condition.

When given inequalities such as \(x > 0\) and \(y < 0\), this means:

• \(x > 0\) suggests choosing values to the right of the y-axis, where x is positive.
• \(y < 0\) limits the area to below the x-axis, where y is negative.

Combining these, we determine the solution set of points is located in Quadrant IV. This process is essential for tasks involving the plotting of inequalities on coordinate planes, ensuring specific conditions are met across a designated region.