Problem 20 - 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. \( x > 4 \)

Step-by-Step Solution

Answer

Under the condition x > 4, (x, y) is located in either Quadrant I or Quadrant IV.

Step 1: Understand the condition

The condition presented is x > 4. This means that the x coordinate must be any value that is greater than 4.

Step 2: Identify the quadrant(s)

Given the condition that x > 4, y can be any value: positive or negative. Therefore, the possible quadrants for (x, y) to be located are Quadrant I (where x and y are both positive) and Quadrant IV (where x is positive, but y is negative).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points and navigate through different quadrants using pairs of numerical coordinates. Each point on this plane is defined by a pair of numbers

• x-coordinate: This indicates horizontal movement, moving right if positive and left if negative.
• y-coordinate: This represents vertical movement, moving up if positive and down if negative.

The plane is divided into four quadrants by the intersection of the x-axis and the y-axis (both coordinate axes): 1. **Quadrant I:** Both x and y are positive. 2. **Quadrant II:** x is negative, y is positive. 3. **Quadrant III:** Both x and y are negative. 4. **Quadrant IV:** x is positive, y is negative. Understanding these quadrants is crucial, especially when working with conditions such as inequalities involving x and y as seen in the exercise.

Inequalities

Inequalities describe the relationship between two expressions that are not necessarily equal and use symbols like <, >, ≤, and ≥ to express this relationship. They can determine the range of possible values for variables on a coordinate plane. For instance, the inequality \[ x > 4 \] tells us that x must be greater than four. This immediately rules out any points where x is less than or equal to four.Applying this to the coordinate plane:

• This inequality means we only focus on those parts of the coordinate plane to the right of the vertical line x = 4.
• Since y is not constrained in the inequality, it can be any positive or negative value.

So, we find that points fulfilling this inequality can be in the first or fourth quadrants because only these quadrants have positive x-values.

Precalculus

Precalculus lays the foundation for understanding concepts in calculus by focusing on functions, analytical geometry, and mathematical modeling. As it introduces students to various mathematical ideas including functions, trigonometry, and limits, it builds critical skills for solving complex calculus problems. When tackling exercises that ask us to determine quadrants for given conditions or inequalities, precalculus teaches us to:

• Interpret and graph inequalities on the coordinate plane effectively.
• Understand the behavior of functions in different quadrants.
• Analyze how restrictions like \[ x > 4 \] limit points to specific quadrants through visualization and graphing.

By gaining proficiency in these areas, students develop the necessary skills for progressing to higher-level mathematics courses seamlessly.