91Ó°ÊÓ

Inequalities help us compare two values or expressions. When we're working with inequalities involving coordinates, the inequality tells us how points relate to each other on the Cartesian plane.
For example, given the inequality \( y < x \), it means the y-coordinate must always be less than the x-coordinate. This is key to understanding where points can or can't lie on the Cartesian plane.
We use inequalities in various real-life scenarios, such as determining budget limits, comparing ages, and much more. Remember, we're not dealing with equal values but with cases where one value is either less than or greater than another. Knowing how to work with inequalities is crucial for solving many mathematical problems.

The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane formed by two perpendicular number lines that intersect at a point called the origin.
It is divided into four sections called quadrants:

• Quadrant I: Both x and y coordinates are positive.


• Quadrant II: The x-coordinate is negative, and the y-coordinate is positive.


• Quadrant III: Both x and y coordinates are negative.


• Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative.

In this exercise, we're focusing on Quadrant II, where x is negative and y is positive. Understanding the layout of the Cartesian plane is essential for graphing equations and inequalities accurately.

Graphing equations involves plotting points that satisfy the equation on the Cartesian plane. When graphing an inequality like \( y < x \), we consider the area of the plane where the inequality holds true.
For this inequality, we need to find all points where the y-coordinate is less than the x-coordinate. This area lies below the line \( y = x \). In Quadrant II, the x-coordinate is always negative, and the y-coordinate is always positive.
Therefore, any positive y-value will always be greater than a negative x-value, making it impossible for \( y < x \) to be true in Quadrant II. Hence, the graph of \( y < x \) cannot lie in Quadrant II. Understanding graphing allows us to visualize and solve inequalities effectively.