Q25 Applications . linear inequalities Homnwor, UnansweredThe final ErjdeE an algebra course based on a #eighted avcrage homework counts 1082, quizzes count 10%e, t...

Solve each problem by using a compound inequality. See Example 12.
Aiming for a $C .$ Professor Johnson gives only a midterm exam and a final exam. The semester average is computed by taking $\frac{1}{3}$ of the midterm exam score plus $\frac{2}{3}$ of the final exam score. To get a C, Beth must have a semester average between 70 and 79 inclusive. If Beth scored only 64 on the midterm, then for what range of scores on the final exam would Beth get a C?

We're going to use a compound inequality to find the range of scores that Jason has to make on his final exam in order to get an average between 70 and 79 inclusive. That last statement that average between 70 and 79 inclusive tells us that we're going to have a less than or equal to for our inequality. The rest of the information Professor Dave counts his midterm as two thirds of the grade means that we're going to add the two thirds of the midterm and then the one third of his final grade in order to get his average grade. So we also know that Jason scored a 64 on the midterm so we can go ahead and substitute that value in for em. So we're going to take two thirds, um, 64 which was his midterm grade. We're gonna add to that one third of his final exam grade will let that be F. And that all has to be between 70 and 79 inclusive. So, looking at the middle part of our compound inequality, if I go ahead and write all of this as, um, two fractions with common denominators I'm gonna have to times 64/3. Plus that one times F or F over three. Finishing out my compound inequality. And I have two fractions in the middle of this with the same denominators. So that tells me that I can add those numerator together. So I'm going toe have two times 64 which is 128 plus f all over three to solve this because my goal is to figure out what that final that final exam grade is. And that's my f I'm going thio try to isolate that f. So I need to first of all, eliminate that three in the denominator. So I'm gonna multiply all three parts of my inequality by three. So 70 times three gives me 210 multiplying and dividing by that same value is gonna cancel that out and give me the 128 plus f and then 79 times three gives me 237 again. I want to get that f all by itself. So I need Thio move that 128 or subtracted from all three parts of my inequality and 210 minus 128 gives me 82. Subtracting out 128 in the middle. Leaves the F by itself and 2 37 minus 128 then is 109. So that means that Jason needs to score between an 82 A 109 on his final exam in order to have an average between 70 and 79 inclusive.

To earn an a. In the course, you must have a final average of at least 90% on the first four examinations. You have grades of 86%, 92% and 84%. If the final examination counts as two grades, what must you get on the final to earn an A. In the course? So an average is when you add up all of your terms and divide by the number of terms there are. Right. So let's start by doing that here. Yeah. So we're just gonna add up all of her grades. So we have 86 plus 88 plus 92 plus 84. Now we also have to include the final exam But that is also two greats. So let's just put that as plus X plus X. Over How many terms are there? There's 1, 345 six terms. And then we're going to let that be equal to 90 Since 90 is what we need to have an eight. Uh huh. So Now let's simplify the numerator here. So let's add up all these terms. So 86 was 88 plus 92 plus 84. That's equal to 350 and X plus X. We can call that two X. So now we have 3 50 plus two X over six is equal to 90. So now we have to solve for X. Yeah. So to solve for X. First we have to get rid of the denominator. So we multiply by six on both sides. So then these cancel out and then we're left with 350 plus two X. Is equal to 540. Now we subtract 3:50 on both sides. Yeah these cancel out. So we're left with two x. Is equal to 5 40 -350. That's 1 90. Okay? And then We divide by two on both sides. So okay X. Is equal to yeah. Yeah 95. So that means in order to get an A. In the class You need a 95 on the final exam. Yeah.

We need to write a system of linear inequalities for the following scenario, we have a smaller truck that can hold £200.1500 cubic feet of cargo. A case of plywood um is 60 cubic feet and weighs £500. A case of tarps is 10 cubic feet and weighs £50 X. Is going to represent the number of cases of plywood and why is going to represent the number of cases of tarps? So we're going to have a pounds equation and then we're going to have a cubic foot equation. So we have £2,000.. So plywood, £500 tarps, £50. And we have exes are plywood and wise are um tarps. So £50 Or £500 times x Plus 50 lb times. Why has to be less than or equal to the £2,000?? Because that's the Mexican hope. Then we have 1500 cubic feet. Plywood is 60 click cubic feet. Then we have 10 cubic feet. So 60 cubic feet. Times X plus 10 cubic feet times Y has to be less than or equal to the 1500 cubic feet. We also have that X. The number of cases of plywood has to be greater than or equal to zero. You can't have negative cases. And why? Same thing? You can have negative cases of tarps? Why has to be greater than or equal to zero as well?

Today, we've been asked to solve two inequalities and find the numbers X that satisfy both of these inequalities are inequalities that we've been given our three X plus four. Listen, are equal to seven and five -2. X is less than soft. Less than 13 soft. Less than means that it's not less than and equal to. So we'll have to pay attention to that when we're defining our set later. So let's just solve these algebraic li like as if these were equal signs. So over here I'm going to get three. X is equal less than or equal to three and then X must be less than or equal to one. And over here I'm going to get five minus 13 which is going to be negative eight. It's going to be less than negative two, X divided by negative two, which means X must be greater than four. So the set of numbers that satisfy both of these inequalities are all numbers where X is bigger than four and X is less than or equal to one. So excesses that are less than or equal to one. Well that's going to go from negative infinity to one hard bracket. And then union soft parentheses E. Four and then all the way up to infinity. Sometimes it's easier to picture this on a number line. So if we we draw this number line here, we're gonna say oh we want all of it. If we put zero here negative 11234 Well it looks like we want all of the numbers that are bigger than four. And I'm gonna just make this look like a better parentheses. So I'm gonna take a parentheses E and draw this arrow shaded in. And then I want all the numbers that are less than or equal to. So I'm going to draw this praise and then shade this arrow in. So we're basically any number that's not in this range in here. Um, because these numbers do not satisfy both inequalities and that's all for today. I hope you join me again next.