Problem 19 - Precalculus with Limits

Question

In Exercises \(15-24,\) determine the quadrant(s) in which \((x, y)\) is located so that the condition(s) is (are) satisfied. $$ y<-5 $$

Step-by-Step Solution

Answer

The point \((x, y)\), for which \(y <-5\), is located in the third and fourth quadrants.

Step 1: Understanding the Layout of the Quadrants

We start by reminding ourselves that the coordinate grid is divided into four quadrants. In the first quadrant, both x and y are positive. In the second quadrant, x is negative and y is positive. In the third quadrant, both x and y are negative. Lastly, in the fourth quadrant, x is positive while y is negative.

Step 2: Analyzing the Condition 'y < -5'

The given condition is that \(y <-5\). This means that the value of y is less than -5. Looking back at the signs of the y-coordinates in all four quadrants, we can see that only in the third and fourth quadrants, y can be negative. Therefore, the condition \(y <-5\) can only occur in the third or the fourth quadrant.

Step 3: Conclusion

In the end, given the condition \(y <-5\), the point \((x, y)\) can only be located in the third and fourth quadrants. This is because only in these quadrants the y-coordinate can be less than -5 as required by the condition.

Key Concepts

Quadrants in the Coordinate System

The coordinate system is divided into four distinct sections known as quadrants. These quadrants help to locate points on a Cartesian plane.
Each quadrant has specific characteristics based on the signs of the x and y coordinates.

• First Quadrant: Both x and y are positive. For example, the point (3, 4) lies here.
• Second Quadrant: x is negative, and y is positive, like at (-2, 5).
• Third Quadrant: Both x and y are negative. For instance, (-6, -3) would be found here.
• Fourth Quadrant: x is positive, but y is negative, such as (4, -7).

By understanding these distinctions, you can determine which quadrant a particular point belongs to by simply examining the signs of its coordinates.

Understanding Inequalities

Inequalities are mathematical expressions that show the relationship between two values where they are not equal. A crucial concept to grasp here is understanding when a value is less than, greater than, or not equal to another value.
Let's break down the inequality from the exercise: - **y < -5**: This means 'y is less than -5.' Therefore, the value of y must be any number that is smaller than -5, such as -6, -10, or -20.
- Inequalities are usually represented on a number line or a graph indicating a range that the variable could fall into. Here, the values of y that meet this condition are all below -5 on a number line.

Understanding inequalities not only helps in solving problems like this one but also forms a critical part of algebraic analysis.

Graphical Analysis of Inequalities

Graphical analysis involves visualizing and understanding mathematical concepts using graphs. In this case, we can graph the inequality to understand where the condition holds true on the coordinate plane. When we say **y < -5**, this implies a horizontal line just below -5 on the y-axis.

• A dotted line is often used to represent the boundary where the inequality changes. Here, y = -5 is this boundary, but it's not included in the solution where y < -5.
• The regions on the graph that satisfy y < -5 are shaded below this line to indicate all points where y fulfills the condition.
• It is critical to recognize, from the general layout of the coordinate system, that these shaded regions exist only in the third and fourth quadrants because that's where y is negative.

By utilizing graphical analysis, students can easily visualize solutions to inequalities and understand in which areas of the graph these solutions apply, making it an invaluable tool in mathematical problem-solving.