Let's do some practice problems dealing with variable expressions. So these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol. So here we have 2 times 11x, so if we have 11 x's and then we're going to have 2 times those 11 x's, we're going to have 22 x's. So another way you could view this, 2 times 11x, you could view this as being equal to 2 times 11, and all of that times x, and that's going to be equal to 22 x's. You had 11 x's, you're going to have 2 times as many x's, so you're going to have 22 x's. Let's see, you have 1.35 times y. Now here we're just going to do a straight simplifying how we write it. So 1.35 times y-- I'll do it in a different color-- 1.35 times y-- that's a little dot there. In algebra we can just get rid of that dot symbol. If we have a variable following a number, we know that means 1.35 times that variable. So that, we could rewrite as just being equal to 1.35y. We've condensed it by getting rid of the multiplication sign. Let's see, here we have 3 times 1/4. Well, this is just straight up multiplying a fraction. So in problem 3-- this was problem 1, this is problem 2, problem 3-- 3 times 1/4, that's the same thing as 3 over 1 times 1/4. Multiply the numerators, you get 3. Multiply the denominators, 1 times 4, you get 4. So number 3, I got 3/4. And then finally, you have 1/4 times z. We could do the exact same thing we did up here in problem number 2. This was the same thing as 1.35y. That's the same thing as 1.35 times y. So down here we could rewrite this as either being equal to 1/4z, or we could view this as being equal to 1 over 4 times z over 1, which is the same thing as z times 1, over 4 times 1, or the same thing as z over 4. So all of these are equivalent. Now, what do they want us to do down here? Evaluate the following expressions for a is equal to 3, b is equal to 2, c is equal to 5, and d is equal to minus 4-- or, actually, I should say negative 4 is the correct terminology. Negative 4. So we just substitute. Every time we see an a, we're going to put a minus 3 there, or a negative 3 there. Every time we see a b, we'll put a positive 2 there. Every time we see a c, we'll put a 5 there. And every time we see a d, we'll put a minus 4 there. And I'll do a couple of these. I won't do all of them, just for the sake of time. So let's say problem number 5. They gave us 2 times a plus 3 times b. Well, this is the same thing as 2 times-- instead of an a, we know that a is going to be equal to negative 3. So 2 times minus 3, plus 3 times b-- what's b? They're telling us that b is equal to 2-- so 3 times 2. And what is this equal to? 2 times minus 3-- let me do it in a different color-- 2 times negative 3 is negative 6, plus 3 times 2. 3 times 2 is 6. That's positive 6. So that is equal to 0. And notice the order of operations. We did the multiplications, we did the two multiplications before we added the two numbers. Multiplication and division takes precedence over addition and subtraction. Let's do problem 6. I'll do that right here. So you have 4 times c. 4 times-- now what's c equal to? They tell us c is equal to 5. So 4 times 5, that's our c, plus d. d is minus or negative 4. So we have 4 times 5 is 20, plus negative 4-- that's the same thing as minus 4, so that is equal to 16. Problem 6. Now, let's do one of the harder ones down here. This problem 10 looks a little bit more daunting. Problem 10 right there. So we have a minus 4b in the numerator, if you can read it, it's kind of small. a is minus 3. So we have minus 3-- or negative 3-- minus 4 times b. b is 2. So 4 times 2. Remember, this right here is a, that right there is b. How do I know? They're telling me up here. And then all of that over-- all of that is over 3c plus 2d. So 3 times-- what was c? c is 5 plus 2 times d. What is d? d is negative 4. So let's figure this out. So we have to do order of operations. Multiplication comes first before addition and subtraction. So this is going to be equal to minus 3 minus 4 times 2, minus 8, all of that over-- 3 times 5 is 15, plus 2 times negative 4 is negative 8, or 15 plus negative 8 is 15 minus 8. And now, the numerator becomes negative 3 minus 8, which is negative 11. And the denominator is 15 minus 8, which is 7. So problem 10, we simplified it to negative 11 over 7. Right there. Let's do a couple of these over here. OK, we see some exponents. I'll pick one of the harder ones. Let's do this one over here, problem 18. So 2x squared minus 3x squared, plus 5x minus 4. OK, well, this wasn't that hard. All of them are dealing with x. But what we could do here-- let me write this down. 2x squared minus 3x squared, plus 5x minus 4. And they tell us that x is equal to negative 1. One thing we could do is simplify this before we even substitute for negative 1. So what's 2 of something minus 3 of something? Right? This is 2x squareds minus 3x squareds. So 2 of something minus 3 of something, that's going to be minus 1 of that something. So that right there-- or negative 1 of that something-- that would be negative 1x squared plus 5x minus 4. And they tell us x is equal to negative 1. So this is negative 1 times x squared, negative 1 squared, plus 5 times x, which is negative 1, minus 4. So what is this? Negative 1 squared is just 1. That's just 1. So this whole expression simplifies to negative 1 plus 5 times negative 1-- we do the multiplication first, of course. So that's minus 5, or negative 5 minus 4. So negative 1 minus 5 is negative 6, minus 4 is equal to negative 10. And I'll do these last two just to get a sample of all of the types of problems in this variable expression section. The weekly cost, c, of manufacturing x remote controls-- so the cost is c, x is the remote controls-- is given by this formula. The cost is equal to 2,000 plus 3 times the number of remote controls, where cost is given in dollars. Question a, what is the cost of producing 1,000 remote controls? Well, the number of remote controls is x. So for part a-- I could write it over here-- the cost is going to be equal to-- just use this formula-- 2,000 plus 3 times the number of remote controls. x is the number of remote controls. So 3 times 1,000. So it's going to be equal to 2,000 plus 3 times 1,000 is 3,000, which is equal to $5,000. So that's part a.$5,000. Now part b. What is the cost of producing 2,000 remote controls? Well, the cost-- just use the same formula-- is equal to 2,000 plus 3 times the number of remote controls. So 3 times 2,000. So that's equal to 2,000 plus 3 times 2,000 is 6,000. So that's equal to \$8,000. Now we're at problem 22. The volume of a box without a lid is given by the formula, volume is equal to 4x times 10 minus x squared, where x is a length in inches, and v is the volume in cubic inches. What is the volume when x is equal to 2? So part a, x is equal to 2. We just substitute 2 wherever we see an x here. So the volume is going to be equal to 4 times x, which is 2, times 10 minus 2 squared. And so this is going to be equal to 4 times 2 is 8, times 10 minus 2 is 8 squared. So that's 8 times-- so this is equal to 8 times-- 8 squared is 64. You could say this is going to be 8 to the third power. And 64 times 8-- 4 times 8 is 32. 6 times 8 is 48, plus 3 is 51. So it's 512. Now, what is the volume when x is equal to 3? I'll do it here in pink. b, when is equal to 3, then the volume is equal to 4 times 3-- x is equal to 3 now-- times 10 minus 3 squared. 4 times 3 is 12, times 10 minus 3 is 7 squared. So it's equal to 12 times 49. And just to get the exact answer, let's multiply that. So 49 times 12-- 2 times 9 is 18, 2 times 4 is 8, plus 1 is 9. Put a 0. 1 times 9 is 9. 1 times 4 is 4. And then we add the two. We get 8, 9 plus 9 is 18, and then 1 plus 4 is 588.