Video Transcript
Fill in the blank: The quadrant representing the solution set of the inequalities ๐ฆ is greater than zero and ๐ฅ is less than zero is the blank quadrant.
Weโre given two different inequalities here: ๐ฆ is greater than zero and ๐ฅ is less than zero. And we need to think about what the region represented by these two inequalities would look like. So letโs sketch out a quick coordinate grid. We know that our ๐ฅ-values go horizontally and our ๐ฆ-values are vertical. And we also know that when we have a coordinate or ordered pair, itโs the ๐ฅ-value followed by the ๐ฆ-value.
So letโs take a look at the first inequality ๐ฆ is greater than zero. The values above the ๐ฅ-axis represent a positive value of ๐ฆ. And the values below the ๐ฅ-axis represent a negative value of ๐ฆ. When ๐ฆ has a positive value, ๐ฆ will be greater than zero, which is the inequality we were looking for. We can therefore eliminate by crossing out the region that we donโt include everything below the ๐ฅ-axis when ๐ฆ is less than zero. We can now include the second inequality where ๐ฅ is less than zero.
We can remember that all the values to the right of the ๐ฆ-axis represent the values when ๐ฅ is a positive value. Everything to the left of the ๐ฆ-axis indicates when ๐ฅ has a negative value. As we want to represent ๐ฅ is less than zero, this will be all the negative values of ๐ฅ. So these will be on the left-hand side. We can eliminate the region where ๐ฅ is greater than zero by crossing it through. So now the region thatโs 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 weโre looking for is this second quadrant. Therefore, the answer to this question is second, since the quadrant representing the solution set of the inequalities ๐ฆ is greater than zero and ๐ฅ is less than zero is the second quadrant.