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## Digital SAT Math

### Course: Digital SAT Math > Unit 2

Lesson 7: Linear inequality word problems: foundations# Systems of linear inequalities word problems — Harder example

Watch Sal work through a harder Systems of linear inequalities word problem.

## Want to join the conversation?

- So ' at least' means 'no less than' and ' equal to or more than'?(71 votes)
- Yes. At least does mean 'greater than or equal to' or 'no less than'. x is
**at least**10, is x ≥ 10(2 votes)

- is there an easy way or one or two clues to mastering this topic? My SAT is next week and I really want to master this but it is kind of hard. Especially ones involving graphs.(19 votes)
- what is the best way to approach and tackle word problems,am poor at it.(10 votes)
- I'd do what Sal does: go and underline the important parts of the word problem. After you find the key parts of the problem, use that simpler information to solve. Hope this helps. :)(33 votes)

- If it said "up to 24" how come you use the greater than/equal to sign? Wouldn't 24 be the maximum? Thanks(8 votes)
- actually, you would use the less than/equal to sign. you might have gotten confused with one of the incorrect answers on the bottom. hope this helped :)(3 votes)

- at1:09

I did not understand why c>or equal to a(4 votes)- Because in the question he said that the one box must have ( at least as many cell phones "c" as accessories "a" ) ..

And ( At least " > " as many " = " ) means > or equal(6 votes)

- At3:11, are you sure that at least 6 means greater than or equal to 6 or less than or equal to 6? Because when I watched it after I worked on the question, I was sure to pick D but was also influenced to pick A.(1 vote)
- At least 6 means that the lowest possible number that our value could be is 6. In other words, the value must be 6 and up, or greater than or equal to 6. It's tough to keep all of the trigger words for inequalities straight sometimes, but "at least" means greater than or equal to and "at most" is less than or equal to.(4 votes)

**Each box must have atleast as many cell phones as accessories**. Can anyone please explain what this line means?(2 votes)- Each box (which is allowed to have 24 units total) must have at least as many cell phones as accessories. This means that there cannot be more accessories than cell phones. The cell phones have to either be the same amount as the accessories or more. This is why C is greater than or equal to A.(2 votes)

- we have mainly been dealing with systems of only two equations. However I have seen some with up to four, would we follow the same steps and continue inserting terms into the following equation?(1 vote)
- When you solve a system of equations by hand, what you are doing is combining equations to eliminate one of the variables. If you use substitution, you solve for the same variable so that you can eliminate it when you set the equations equal to each other. If using elimination, you multiply the equations so that when you add (combine) them, one variable is eliminated.

The same process works for multi-variable systems (although I'm pretty sure those won't ever show up on the SAT). If, say you were given three variables, you would take two of the equations, and cancel out a variable. Then, you could take a different pair of equations, and combine them in such a way that you cancel out the same variable. Then you just have a normal 2-equation 2-variable system, which you can solve. Lastly, you plug in your answers for those two variables that you just found into one of the original variables to get the third missing one back. The same general procedure can be used for four variables, although it gets more tedious every time.(4 votes)

- the last two options are the same. cellphones are as much as accesories and accessories are as much as cellphones but why option c??....cant we choose option d?(2 votes)
- Option C says a is less than/equal to c, which is what we want. Option D says c is less than/equal to a. There can be multiple parts to an answer, so be careful. Remember: If it's half-wrong, then it's all wrong! (meaning don't choose it if it only has half of the correct information)(1 vote)

## Video transcript

- [Instructor] We're told,
"Luis is cooking meals for at least 20 people. He estimates that the cost of
each vegetarian meal is $3, and the cost of each
meal with meat is $4.50. His budget for the meals
is no more than $100, and he wants to cook at least
six of each type of meal. Which of the following systems
of inequalities represents the conditions described if x is the number of vegetarian meals and y is the number of
meals with meat Luis cooks?" So pause this video and
have a go at this yourself before we work through it together. And I know this is a long question and these systems feel
complicated, but trust me, if you do it step-by-step
you'll actually find that it all falls into place. All right, now let's
work through it together, and it's important to emphasize that they've already defined
the two key variables for us. x is the number of vegetarian meals and y is the number of meals with meat. So let's look at each of
the constraints they give us and each of these can set
up a different inequality. So the first one is, they
say, "Luis is cooking meals for at least 20 people." So that tells us that the
total number of meals, which is going to be the
number of vegetarian meals, that's x, plus the number of meat meals, that has to be at least equal to 20. So that has to be greater
than or equal to 20. That's what that first sentence tells us. And if I wasn't doing
this as a multiple choice, I would just keep adding more
and more constraints here, but they give us some choices. And so we can see that x plus
y is greater than or equal to 20, that's in choice A. It's actually not in choice B. So we can already rule out choice B. They have less than or equal to 20 here. Same thing for choice C.
So we can rule that out. And then choice D does have that. So we are still in the running. The next constraint they tell us, "He estimates that the cost
of each vegetarian meal is $3, and the cost of each
meal with meat is 4.50. His budget for the meals
is no more than $100." So how much is he gonna spend in total? Well, on the vegetarian meals, he's going to spend the
number of vegetarian meals times $3 per meal. So that's how much he's going
to spend on vegetarian meals. And what about meat meals? Well, it's going to be y
meals times 4.50 per meal. So it's 4.5 times y. The amount that he's
spending on vegetarian meals, the amount that he's spending
on non-vegetarian meals, that's the total he's spending on meals. And they say it is no more than $100. So this has to be less
than or equal to 100. And so let's see, we actually over here, we have 3x plus 4.5y is
greater than or equal to 100. So we can rule choice A out as well and just by deductive reasoning, we see choice D does have that in there. But this must be the answer, but let's keep going to make sure that these other constraints work. We are also told he wants to cook at least six type of each meal. So that means that x, the
number of vegetarian meals, has to be greater than or equal to six. And y, the number of non-vegetarian meals, also has to be greater
than or equal to six. And we see both of these down here, so we can feel pretty good about choice D.