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Graphs of systems of inequalities word problem

Given the graph of a two-variable linear inequality that models a context about chopping vegetables, Sal finds if there's enough time to chop!

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• At it is said carrots and broccoli take the same time to get cut but graph of inequality doesn't show it. It says 15 broccoli and 0 carrots can be cut in 540 seconds at the same time shows 30 carrots and 0 broccoli can be cut in 540 seconds.
Does this mean,
15 B + 0 C = 540 s
30 C + 0 B = 540 s

B= 36 s
C= 18 s
??
This contradicts the fact in the question
(1 vote)
• It actually says that it takes the same number of seconds to chop each carrot and the same number of seconds to chop each broccoli. This does not mean that the carrots and broccoli are chopped at the same rate, but that the rate of chopping carrots is constant (as you found 18 sec/carrot) and the rate of chopping broccoli is constant (36 sec/carrot).
(17 votes)
• Does this mean that carrots are cut at a rate of 540/30 and broccoli heads are cut at a rate of 540/15?
(8 votes)
• Yes, although it probably wouldn't be literal in real life.
(0 votes)
• Q: How do you know which variable to assign to which axes.
I saw that the y axes was replaced by B and the x axes by C.
Why not have C be the y axes and B be the x axes?
(0 votes)
• They both work! The nice thing about algebra is that you can make the axes whatever you want. Sometimes there are conventions (time is the x axis), but it doesn't matter which you use.
(9 votes)
• Can anyone give me some context on this problem like what sal is tryna to explain?
(0 votes)
• Can someone plz help me on how the answer is 20? At , Why does Sal shade the overlap?
(0 votes)
• sal shades the overlap due to the property of inequalitites. when inequalitites are graphed and multiple inequalities are graphed, to find answers that satisfy all of the inequaitites you will need to shade in a particular part of the graph. to help remember how to graph these inequalities, remember:
< is a dashed line and shading below
> is a dashed line and shading above
≤ is a solid line and shading below
≥ is a solid line and shading above
(5 votes)
• Could we have calculated equation B from the information given, or not? I noticed that it was given a y-int of 15 and x-int of 30 - this must have been using additional information? If not, I would appreciate any advice on how it could be calculated from the info given. Thanks.
(0 votes)
• Will it always be the intersection point?
(0 votes)
• Anoosha,

No, there will not always be an intersection point. There are times when the lines will be parallel, which makes the answer the empty set.
(0 votes)
• What information was used to draw the line through (0,15) and (30,0)?
(0 votes)
• this still makes no sense
(0 votes)
• Why is it shaded blue if it says she wants to chop at least 20 vegetales? Should it not be below?
(0 votes)
• Isaiah,

Think of it this way:

She wants to chop at least 20 vegetables - NOT less than 20 vegetables.

She wants to chop 20 vegetables or more.

Vegetables she wants to chop >= 20.
(6 votes)

Video transcript

- [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.