Graphs of systems of inequalities word problem

Ksenia wants to chop broccoli and carrots for a competition it takes your 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 five hundred and forty 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 senia 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 of this blue shaded area and even the line is a solid line so it includes point of the line these are all of the scenarios where she's chopping at least 20 vegetables all this blue 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 this is all of the combinations where she is within her time limit where she's 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 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 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 on the Green Line and so the overlap of the two if she's meeting both constraints it's going to be it's going to be all of all of this area all of this area right over here this is the overlap this is the overlap of the two of the two solution sets so in this overlap where is the least number of carrots what is the least number of carrots and you can chop while achieving her goal so if we see here the least number of cats you might be the attempt 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 you can actually find a combination that has even fewer carrots you could 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 10 carrots and 10 broccoli 10 broccoli heads broccoli 10 and broccoli broccoli heads or 10 let me just write that 10 broccoli heads so that's the least amount any if you wanted to somehow figure out less than 10 carrots in any of those scenarios there's no overlap you know if you said that is there any way to do 9 carrots if you look over here there's no overlap at C equals 9 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 while achieving frankly her goals both of her goals being under time and chopping at least 20 vegetables