Question Video: Locating Solution Sets of Inequalities in the Cartesian Plane Mathematics

Video Transcript

Fill in the blank: The quadrant representing the solution set of the inequalities 𝑦 is less than zero and 𝑥 is greater than zero is the what quadrant.

We recall that the coordinate plane has an 𝑥-axis that divide it into a top and bottom half and a 𝑦-axis dividing it into a left and right half. Together they create the four quadrants of the plane as shown. By convention, these are often numbered using Roman numerals. Starting from the positive 𝑥-axis, we travel in a counterclockwise direction to reach the second, third, and fourth quadrants, respectively. The first quadrant is in the top right, the second in the top left, the third in the bottom left, and the fourth quadrant is the bottom right.

In this question, we are given two inequalities, 𝑦 is less than zero and 𝑥 is greater than zero. If 𝑦 is less than zero, it must be negative. Therefore, our 𝑦-value must lie below the 𝑥-axis. This means it lies in either the third or fourth quadrants. As we are also told that 𝑥 is greater than zero, our 𝑥-value must be positive. This means it must lie to the right of the 𝑦-axis. If 𝑥 is greater than zero, we are in either the first or fourth quadrants.

As our solution set needs to satisfy both inequalities, we must be in the fourth quadrant. If 𝑦 is less than zero, we cannot be in the first quadrant. Likewise, if 𝑥 is greater than zero, we cannot be in the third quadrant. The quadrant representing the solution set of the inequalities 𝑦 is less than zero and 𝑥 is greater than zero is the fourth quadrant.