Fill in the blank: the quadrant representing the solution set of the inequalities 𝑦 is less than zero and 𝑥 is less than zero is the what quadrant.
We begin by recalling that the coordinate plane has an 𝑥-axis that divides it into a top and bottom half and the 𝑦-axis, dividing it into the left and right half. Together, they create the four quadrants of the plane. By convention, these are often numbered using Roman numerals from I to IV. The first quadrant is the top right. And going in a counterclockwise direction from the positive 𝑥-axis, we see that the second, third, and fourth quadrants are as shown. The second quadrant is the top left, the third the bottom left, and the fourth the bottom right.
In this question, we’re given two inequalities: 𝑦 is less than zero and 𝑥 is less than zero. If 𝑦 is less than zero, the 𝑦-coordinate of any point in the solution set must lie below the 𝑥-axis as 𝑦 is negative. This means that our point must lie in the third or fourth quadrant. We are also told that 𝑥 is less than zero, which means that the 𝑥-coordinate must be negative. This means it lies to the left of the 𝑦-axis and must be in either the second or third quadrant. The solution set has to satisfy both inequalities. Therefore, our point must lie in the third quadrant.
The quadrant representing the solution set of the inequalities 𝑦 is less than zero and 𝑥 is less than zero is the third quadrant.