Problem 21 - 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 point \((x, y)\) satisfying the conditions \(x < 0\) and \(-y > 0\) is located in the 3rd quadrant of the Cartesian coordinate system.

Step 1: Analyze the condition \(x < 0\)

This condition tells that the x-coordinate of the point is less than zero i.e., x is negative. In a Cartesian coordinate system, if x is negative, then the point either lies in the 2nd or the 3rd quadrant.

Step 2: Analyze the condition \(-y > 0\)

This condition tells that the negative of y-coordinate is greater than zero i.e., y is negative. In the Cartesian system, if y is negative, the point either lies in the 3rd or 4th quadrant.

Step 3: Determine the common quadrant

To satisfy both conditions, the point must be in a quadrant where both x and y are negative. The only quadrant that satisfies these criteria is the 3rd quadrant.

Key Concepts

Quadrants

In the Cartesian coordinate system, the plane is divided into four quadrants by the x-axis and y-axis. These quadrants are numbered in a counterclockwise direction starting from the top right. Understanding which quadrant a point is in helps us interpret the signs of its coordinates.

• 1st Quadrant: Both x and y coordinates are positive.
• 2nd Quadrant: x is negative, and y is positive.
• 3rd Quadrant: Both x and y coordinates are negative.
• 4th Quadrant: x is positive, and y is negative.

The position of a point, defined by its coordinates, determines in which quadrant the point is located.

Inequalities

Inequalities allow us to express the relative size of numbers. In this context, they are used to describe the conditions for the coordinates of a point. When we have an inequality such as \(x < 0\), we know that x is on the left side of the y-axis. Similarly, an inequality like \(-y > 0\) suggests a condition for y. Let's break them down:

• \(x < 0\): The x-coordinate is less than zero, hence in the negative region.
• \(-y > 0\): Rewriting this as \(y < 0\), it tells us that y is negative.

These inequalities help determine which quadrant a point is in by specifying the sign of the coordinates.

Coordinates

Coordinates are an essential component of the Cartesian coordinate system. They are used to pinpoint the location of a point on the plane. Each coordinate pair \((x, y)\) consists of two numbers which indicate the position relative to the x-axis and y-axis. These coordinates are written as ordered pairs:

• x-coordinate: Determines how far left or right a point is from the origin.
• y-coordinate: Indicates how far up or down a point is from the origin.

The pair \((x, y)\) enables us to unambiguously locate any point in the plane by using these two critical values.

X-coordinate

The x-coordinate tells us the horizontal position of a point on the Cartesian plane. It measures the distance of a point from the y-axis and can be positive, negative, or even zero. A few key points to remember:

• Positive x-coordinate: Points are located to the right of the y-axis.
• Negative x-coordinate: Points are positioned to the left of the y-axis.
• x = 0: The point is on the y-axis itself.

In the given problem, the condition \(x < 0\) tells us that the point is to the left of the y-axis, limiting its position to either the 2nd or 3rd quadrant.

Y-coordinate

The y-coordinate is crucial in defining the vertical position of a point within the Cartesian plane. It determines how a point is positioned relative to the x-axis and can also be positive, negative, or zero.

• Positive y-coordinate: The point lies above the x-axis.
• Negative y-coordinate: The point is below the x-axis.
• y = 0: The point is on the x-axis.

In the exercise, since \(-y > 0\) translates to \(y < 0\), it tells us that the point is positioned below the x-axis, and thus could be either in the 3rd or 4th quadrant. Together with the condition for the x-coordinate, it determines the correct quadrant as the 3rd.