In Exercises 75–78, list the quadrant or quadrants satisfying eac...

Question

In Exercises 75–78, list the quadrant or quadrants satisfying each condition.

x^3 > 0 and y^3 <0

Accepted Answer

Welcome back. I'm so glad you're here. We are given two conditions. We're told that one minus X cubed is greater than zero and we're told that one minus Y cubed is less than zero and we are to identify the quadrants where those conditions form a region. Well when we've got a coordinate plane with a horizontal X axis and a vertical Y axis, they form four quadrants. The the quadrant up to the right where X and Y are both positive is quadrant one to the left of that where X is negative and Y is positive is quadrant two to the lower left of that where they're both X and y are both negative is quadrant three and the only one left where X is positive and why is negative is quadrant four. And we now want to graph these two inequalities, one minus X cubed is greater than zero and one minus Y cubed is less than zero. And we're going to find what regions what quadrants are contained in those regions. So the first thing we want to do actually is to change this inequality into a line and graph our lines. So for a line we would set it equal to zero and then solve for X. We're going to subtract one from both sides negative X cubed equals negative one, divide both sides by negative one, X cubed equals one. Take the cubed root of both sides and the cubed root of one is one. So X equals one, which means we could have any value for Y but X is always going to be one. So starting from the origin, we move to the right one. We find one on the X axis and draw a vertical line. And now we want to find out if we're talking about when we have our inequality one minus X cubed is greater than zero. Are we talking about the region here? To the right of the line, X equals one or the region here to the left of the line where X equals one. So what we do is we choose a test point and when it's available meaning when it's not part of your line, the best test point is the origin 00 because we're just going to stick a zero in for X and that's not too tricky to solve zero cubed. That's just zero. So we've got one is greater than zero. Is that true? Is one greater than zero? Yes it is. So we're talking about the region that contains our test point. We're talking about this region here. And actually that region from this first inequality, it contains all four quadrants. So that didn't really help us much. Let's hope that our Y one minus Y cubed is less than zero. Will help us out a little bit more. Again, let's graph our line first will set that equal to zero. Subtract one from both sides, negative Y cubed equals negative one, divide both sides by negative one. We've got Y cubed equals one. Take the cubed root of both sides. And we've got Y equals while the cubed root of one is still one. Why equals one. So that means X could be any value up to a to Gillian and why is still going to be one? So from the origin we go up one and then we draw a horizontal line For y equals one. And now we want to know are we talking about the region below our line, Y equals one or the region above our line Y equals one. And to do that we again choose a test 10.0 is still available to us the origin. So we will stick a zero in for Y. One minus zero cubed is less than 00 cubed. Still nothing. So one is less than zero. Is that true? No one's not less than zero. So this time it doesn't it's the region that does not contain our test point that does not contain the origin or zero. So we're talking about this area up above our line. Why equals one. And now we want to look for the part that contains both the blue and the green shade that contains shade the regions from both of our inequalities. And that is going to be this section to the left of our line. Y equals one and above our line. Or excuse me? To the left of our line X equals one and above our line Y equals one. And that is some of the area in both regions one and two. You can see this narrow area here from quadrant one. Excuse me? And most of quadrant two. So we look at our answer choices and that matches with answer choice D. Well done, We'll catch you on the next one.

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