Graphs of systems of inequalities word problem

- [Voiceover] "Ksenia wants to chop broccoli "and carrots for a competition. "It takes her the same number of seconds "to chop each carrot, and it takes her the same number "of seconds to chop each broccoli head. "Her goal is to chop at least 20 vegetables "with a time limit of 540 seconds," all right. "The graph below represents the set "of all combinations of carrots and broccoli. "Inequality A," let's see, "Inequality A represents "the range of all combinations Ksenia wants to chop." Because she wants to chop at least 20 vegetables. So, that's what Inequality A is representing, that she wants to chop at least 20 vegetables. So, all this blue shaded area and even the line, is a solid line so it includes point on the line. These are all of the scenarios where she's chopping at least 20 vegetables, all this blue area including the blue line. And it says, "Inequality B represents the range of all combinations she can chop with her time limit." So, Inequality B, this is all of the combinations where she is within her time limit, where she is not spending any more than 540 seconds. "What is the least number of carrots Ksenia can chop while achieving her goal?" Well, her goal, remember she wants to chop at least 20 vegetables. So, you want to be in the blue area. You want to be in the solution set for Inequality A, which would be the blue area or on the blue line. And she wants to achieve her goal of meeting the time limit. So, she needs to also be in the solution set for Inequality B so she also has to be in the green area or on the green line. And so the overlap of the two, if she's meeting both constraints, it's going to be all of this area. This is the overlap of the two solution sets. So, in this overlap where is the least number of carrots. "What is the least number of carrots Ksenia can chop while achieving her goal." So, if we see here, the least number of carrots, you might tempted to say, "Okay, 20 carrots, that is in the solution set." That would be 20 carrots and zero broccoli heads but you can actually find a combination that has even fewer carrots. You can go all the way to this point because remember the points on the lines are also included in the solution sets, because they are solid lines not dash lines. So, this point right over here, 10 carrots and 10 broccoli heads actually meets her goal. So, let me write that down. 10 carrots and 10 broccoli, 10 broccoli heads. Let me just write that, 10 broccoli heads. So, that's the least amount. If you wanted to somehow figure out less than 10 carrots, in any of those scenarios there's no overlap. You know, if you say, "Oh, is there any way "to do nine carrots?" If you look over here there's no overlap at c equals nine between the two solution sets. So, the minimum right over here is actually the point of intersection of these two lines. 10 carrots, 10 broccoli heads that's the combination that has her chopping the minimum number of carrots while achieving, frankly, her goals, both of her goals. Being under time and chopping at least 20 vegetables.