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### Course: Algebra 1 > Unit 7

Lesson 3: Modeling with linear inequalities- Writing two-variable inequalities word problem
- Solving two-variable inequalities word problem
- Interpreting two-variable inequalities word problem
- Two-variable inequalities word problems
- Modeling with systems of inequalities
- Writing systems of inequalities word problem
- Solving systems of inequalities word problem
- Graphs of systems of inequalities word problem
- Systems of inequalities word problems
- Graphs of two-variable inequalities word problem
- Inequalities (systems & graphs): FAQ
- Creativity break: What can we do to expand our creative skills?

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# Writing systems of inequalities word problem

Sal models a word problem about baking cupcakes and muffins by creating a system of linear inequalities.

## Want to join the conversation?

- I have no idea what i am doing!(16 votes)
- I know that feeling. Try to understand what he is saying intuitively. Then it will come out faster than the alphabet.(2 votes)

- Does 'At most' always refer to 'Less than or equal to'(6 votes)
- not always but yes .(3 votes)

- If you graph them, do you get the "feasible region" in other problems?(4 votes)
- If there is/are solutions that satisfy both conditions, then yes, when you plot it, it will show the feasible regions. If there is no true answer then no, there will not be a feasible region.(4 votes)

- what does at least stand for? like what symbol?(1 vote)
- "at least" is a value that you can't go below. If you need at least $500 each week to pay your bills, then you need ">=500"

"at most" is a value that you can't go above. If you want to spend at most $100 on groceries each week, then you can spend "<=100".

Hope this helps.(7 votes)

- Can you write an inequality that represents the amount of cupcakes made or muffins made? Or is the information insufficient/impossible to solve for?(4 votes)
- I don't think so.

We have in the first sentence an inequality based on sugar and then in the second inequality we have all setup based on flour.

Even if we have the same variable, C and M their constants and meaning aren't the same.(3 votes)

- Can this system of inequalities be solved?(2 votes)
- it is solvable, but the big numbers might make it harder to solve unless you use a graphing calculator such as DESMOS which is a free on-line calculator. This will show a double shaded region with possible solutions, but you can only use the whole number solutions, it looks like somewhere between 40 to 50 possible solutions.(3 votes)

- Just want to confirm something, so in inequality the solution is not just one, but many as long as in shaded area right?(3 votes)
- How would you solve this system of inequalities?

Would elimination/substitution work?(2 votes)- Yeah you could use both. They're both compatible(1 vote)

- to think I forgot what at most and at least even meant(1 vote)
- "At most" means that it can be less than or equal to. "At least" means that it can be greater than or equal to.(3 votes)

- a number no more than -8(0 votes)
- The words "no more than" mean that you can't go higher than -8. Or, your value needs to be smaller than or equal to -8.

So, your words create: x <= -8(2 votes)

## Video transcript

- [Voiceover] So we're told
a cupcake requires 35 grams of sugar and 50 grams of flour. So let me underline that. "A cupcake requires 35 grams of sugar "and 50 grams of flour." And then they tell us, "A muffin requires 30 grams of sugar." So let me underline that. "A muffin requires 30 grams of sugar and 65 grams of flour." All right, so what are they
going to ask us to actually do? "Shawna needs to use up at
least 460 grams of sugar." So let me underline
this in different color. She needs to use up, "Shawna needs to use up at
least 460 grams of sugar "to make cupcakes and muffins, "and she wants to use at
most 970 grams of flour." So let me use another color. "And she wants to use "at most 970 grams of flour. "Let's form a system of inequalities "to represent Shawna's conditions. "Let C denote the number
of cupcakes she makes "and M the number of muffins she makes." All right, "Write an
inequality that represents "the condition based on the
number of grams of sugar." So the condition based on
the number of grams of sugar. That's what we underlined here in green. "Shawna needs to use up at
least 460 grams of sugar "to make cupcakes and muffins." So let's see how we can write that out. So how much sugar she's
gonna use for the cupcakes? Well, we're told right over here she requires 35 grams
of sugar per cupcake. So the amount of sugar
that she's going to use for all of the cupcakes is going to be 35 grams per cupcake, let me write that a little bit neater, 35 grams per cupcake times
the number of cupcakes. So this is how much sugar total she's going to use on the cupcakes. 35 grams per cupcake times the number of cupcakes. And then how many grams of sugar is she going to use for the muffins? Well, that's going to be 30
grams of sugar per muffin. So plus 30 grams of sugar per muffin times the number of muffins. So times M. So this is the total
sugar from the cupcakes. This is the total sugar from the muffins. And so if you add them together, this is a total sugar that she's using and we're told that she needs to use up at least 460 grams of sugar. So everything here is in grams. So this has to be greater than or equal. She needs to use at
least 460 grams of sugar to make both of these. So that's it. We can just set up the
inequality that represents the condition based on the
number of grams of sugar which is this condition right over here. Now let's see, this next question. Let's see, "Write an
inequality that represents "the condition based on the
number of grams of flour." Well, that's what we underlined
in orange right over here. She needs to use no at
most 970 grams of flour. So no more than 970 grams. So how would we express
that as an inequality? Well, how much flour is she
going to use for the cupcakes? Well, she used 50 grams
of flour per cupcake. So 50 grams per cupcake
times the number of cupcakes. That's how much flour she's
going to use for the cupcakes. And then for the muffins, she uses 65 grams of flour per muffin times the number of muffins. So this is the total amount of flour that she is going to use and this has to be less
than or equal to 970 grams. Less than or equal to 970 and we're done. And so you could actually,
if you wanted to, you could say, "What pairs of C and M "satisfy both of these inequalities?" And then you could say,
"Okay, that's the combination "of the number of cupcakes
and muffins she can make "in order to satisfy
her actual conditions."