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Let's do some practice problems dealing with variable

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expressions.

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So these first problems say write the following in a more

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condensed form by leaving out the multiplication symbol or

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leaving out a multiplication symbol.

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So here we have 2 times 11x, so if we have 11 x's and then

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we're going to have 2 times those 11 x's, we're going to

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have 22 x's.

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So another way you could view this, 2 times 11x, you could

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view this as being equal to 2 times 11, and all of that

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times x, and that's going to be equal to 22 x's.

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You had 11 x's, you're going to have 2 times as many x's,

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so you're going to have 22 x's.

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Let's see, you have 1.35 times y.

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Now here we're just going to do a straight simplifying how

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we write it.

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So 1.35 times y-- I'll do it in a different color-- 1.35

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times y-- that's a little dot there.

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In algebra we can just get rid of that dot symbol.

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If we have a variable following a number, we know

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that means 1.35 times that variable.

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So that, we could rewrite as just being equal to 1.35y.

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We've condensed it by getting rid of the

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multiplication sign.

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Let's see, here we have 3 times 1/4.

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Well, this is just straight up multiplying a fraction.

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So in problem 3-- this was problem 1, this is problem 2,

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problem 3-- 3 times 1/4, that's the same thing as 3

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over 1 times 1/4.

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Multiply the numerators, you get 3.

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Multiply the denominators, 1 times 4, you get 4.

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So number 3, I got 3/4.

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And then finally, you have 1/4 times z.

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We could do the exact same thing we did up here in

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problem number 2.

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This was the same thing as 1.35y.

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That's the same thing as 1.35 times y.

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So down here we could rewrite this as either being equal to

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1/4z, or we could view this as being equal to 1 over 4 times

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z over 1, which is the same thing as z times 1, over 4

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times 1, or the same thing as z over 4.

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So all of these are equivalent.

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Now, what do they want us to do down here?

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Evaluate the following expressions for a is equal to

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3, b is equal to 2, c is equal to 5, and d is equal to minus

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4-- or, actually, I should say negative 4 is the correct

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terminology.

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Negative 4.

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So we just substitute.

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Every time we see an a, we're going to put a minus 3 there,

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or a negative 3 there.

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Every time we see a b, we'll put a positive 2 there.

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Every time we see a c, we'll put a 5 there.

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And every time we see a d, we'll put a minus 4 there.

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And I'll do a couple of these.

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I won't do all of them, just for the sake of time.

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So let's say problem number 5.

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They gave us 2 times a plus 3 times b.

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Well, this is the same thing as 2 times-- instead of an a,

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we know that a is going to be equal to negative 3.

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So 2 times minus 3, plus 3 times b-- what's b?

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They're telling us that b is equal to 2-- so 3 times 2.

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And what is this equal to?

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2 times minus 3-- let me do it in a different color-- 2 times

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negative 3 is negative 6, plus 3 times 2.

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3 times 2 is 6.

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That's positive 6.

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So that is equal to 0.

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And notice the order of operations.

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We did the multiplications, we did the two multiplications

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before we added the two numbers.

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Multiplication and division takes precedence over addition

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and subtraction.

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Let's do problem 6.

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I'll do that right here.

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So you have 4 times c.

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4 times-- now what's c equal to?

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They tell us c is equal to 5.

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So 4 times 5, that's our c, plus d.

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d is minus or negative 4.

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So we have 4 times 5 is 20, plus negative 4-- that's the

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same thing as minus 4, so that is equal to 16.

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Problem 6.

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Now, let's do one of the harder ones down here.

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This problem 10 looks a little bit more daunting.

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Problem 10 right there.

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So we have a minus 4b in the numerator, if you can read it,

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it's kind of small.

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a is minus 3.

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So we have minus 3-- or negative 3-- minus 4 times b.

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b is 2.

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So 4 times 2.

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Remember, this right here is a, that right there is b.

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How do I know?

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They're telling me up here.

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And then all of that over-- all of that is

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over 3c plus 2d.

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So 3 times-- what was c? c is 5 plus 2 times d.

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What is d?

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d is negative 4.

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So let's figure this out.

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So we have to do order of operations.

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Multiplication comes first before addition and

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subtraction.

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So this is going to be equal to minus 3 minus 4 times 2,

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minus 8, all of that over-- 3 times 5 is 15, plus 2 times

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negative 4 is negative 8, or 15 plus negative

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8 is 15 minus 8.

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And now, the numerator becomes negative 3 minus 8, which is

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negative 11.

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And the denominator is 15 minus 8, which is 7.

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So problem 10, we simplified it to negative 11 over 7.

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Right there.

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Let's do a couple of these over here.

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OK, we see some exponents.

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I'll pick one of the harder ones.

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Let's do this one over here, problem 18.

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So 2x squared minus 3x squared, plus 5x minus 4.

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OK, well, this wasn't that hard.

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All of them are dealing with x.

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But what we could do here-- let me write this down.

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2x squared minus 3x squared, plus 5x minus 4.

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And they tell us that x is equal to negative 1.

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One thing we could do is simplify this before we even

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substitute for negative 1.

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So what's 2 of something minus 3 of something?

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Right?

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This is 2x squareds minus 3x squareds.

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So 2 of something minus 3 of something, that's going to be

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minus 1 of that something.

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So that right there-- or negative 1 of that something--

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that would be negative 1x squared plus 5x minus 4.

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And they tell us x is equal to negative 1.

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So this is negative 1 times x squared, negative 1 squared,

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plus 5 times x, which is negative 1, minus 4.

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So what is this?

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Negative 1 squared is just 1.

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That's just 1.

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So this whole expression simplifies to negative 1 plus

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5 times negative 1-- we do the

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multiplication first, of course.

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So that's minus 5, or negative 5 minus 4.

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So negative 1 minus 5 is negative 6, minus 4 is equal

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to negative 10.

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And I'll do these last two just to get a sample of all of

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the types of problems in this variable expression section.

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The weekly cost, c, of manufacturing x remote

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controls-- so the cost is c, x is the remote controls-- is

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given by this formula.

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The cost is equal to 2,000 plus 3 times the number of

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remote controls, where cost is given in dollars.

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Question a, what is the cost of producing

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1,000 remote controls?

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Well, the number of remote controls is x.

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So for part a-- I could write it over here-- the cost is

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going to be equal to-- just use this formula-- 2,000 plus

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3 times the number of remote controls.

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x is the number of remote controls.

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So 3 times 1,000.

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So it's going to be equal to 2,000 plus 3 times 1,000 is

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3,000, which is equal to $5,000.

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So that's part a. $5,000.

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Now part b.

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What is the cost of producing 2,000 remote controls?

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Well, the cost-- just use the same formula-- is equal to

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2,000 plus 3 times the number of remote controls.

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So 3 times 2,000.

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So that's equal to 2,000 plus 3 times 2,000 is 6,000.

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So that's equal to $8,000.

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Now we're at problem 22.

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The volume of a box without a lid is given by the formula,

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volume is equal to 4x times 10 minus x squared, where x is a

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length in inches, and v is the volume in cubic inches.

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What is the volume when x is equal to 2?

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So part a, x is equal to 2.

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We just substitute 2 wherever we see an x here.

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So the volume is going to be equal to 4 times x, which is

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2, times 10 minus 2 squared.

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And so this is going to be equal to 4 times 2 is 8, times

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10 minus 2 is 8 squared.

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So that's 8 times-- so this is equal to 8 times--

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8 squared is 64.

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You could say this is going to be 8 to the third power.

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And 64 times 8-- 4 times 8 is 32.

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6 times 8 is 48, plus 3 is 51.

189
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So it's 512.

190
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Now, what is the volume when x is equal to 3?

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I'll do it here in pink. b, when is equal to 3, then the

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volume is equal to 4 times 3-- x is equal to 3 now-- times 10

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minus 3 squared.

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4 times 3 is 12, times 10 minus 3 is 7 squared.

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So it's equal to 12 times 49.

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And just to get the exact answer, let's multiply that.

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So 49 times 12-- 2 times 9 is 18, 2 times 4 is

198
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8, plus 1 is 9.

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Put a 0.

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1 times 9 is 9.

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1 times 4 is 4.

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And then we add the two.

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We get 8, 9 plus 9 is 18, and then 1 plus 4 is 588.

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