Fill in the blank: The quadrant representing the solution set of the inequalities 𝑦 is greater than zero and 𝑥 is less than zero is the blank quadrant.
We’re given two different inequalities here: 𝑦 is greater than zero and 𝑥 is less than zero. And we need to think about what the region represented by these two inequalities would look like. So let’s sketch out a quick coordinate grid. We know that our 𝑥-values go horizontally and our 𝑦-values are vertical. And we also know that when we have a coordinate or ordered pair, it’s the 𝑥-value followed by the 𝑦-value.
So let’s take a look at the first inequality 𝑦 is greater than zero. The values above the 𝑥-axis represent a positive value of 𝑦. And the values below the 𝑥-axis represent a negative value of 𝑦. When 𝑦 has a positive value, 𝑦 will be greater than zero, which is the inequality we were looking for. We can therefore eliminate by crossing out the region that we don’t include everything below the 𝑥-axis when 𝑦 is less than zero. We can now include the second inequality where 𝑥 is less than zero.
We can remember that all the values to the right of the 𝑦-axis represent the values when 𝑥 is a positive value. Everything to the left of the 𝑦-axis indicates when 𝑥 has a negative value. As we want to represent 𝑥 is less than zero, this will be all the negative values of 𝑥. So these will be on the left-hand side. We can eliminate the region where 𝑥 is greater than zero by crossing it through. So now the region that’s represented by both inequalities is here in the top left. The question is, which quadrant is this?
When the quadrants are numbered, we start with the top right as the first quadrant and then work our way round counterclockwise to the fourth quadrant. The region that we’re looking for is this second quadrant. Therefore, the answer to this question is second, since the quadrant representing the solution set of the inequalities 𝑦 is greater than zero and 𝑥 is less than zero is the second quadrant.