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## Modeling with linear inequalities

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# Writing systems of inequalities word problem

CCSS Math: HSA.CED.A.3

## 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."