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In $21-24,$ the length and width of a rectangle are expressed in terms of a variable.a. Express each perimeter in terms of the variable.b. Express each area in terms of the variable.$$l=2 x \text { and } w=\frac{1}{x}$$

a. $\frac{4 x^{2}+2}{2}$b. 2

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 4

Adding and Subtracting Rational Expressions

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

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We're told that we were dealing with the rectangle, and we're told that the length of that, which would be the longer sides of the rectangle, the length of this rectangle is two X. And we're told that the width of this rectangle, the shorter sides was one over X and then were asked to find both the perimeter and the area times. We'll start with perimeter. As you guys probably already know. The way you find the perimeter of a rectangle is you just simply take the links of all of its sides and added up, meaning you're taking length plus length plus with plus with Okay, So if you want to write that out length plus length, plus with plus with that's what you're trying to do here. So in this case, that would mean for perimeter. You are doing your two lengths of two X and two X, and you are adding that to your two wits of one over X and one over X so immediately, I see that I have two terms that are listed as fractions in two terms that are not well as we've talked about. I prefer to have either everything a fraction or everything. The whole numbers. Since I can't change the fractions, the whole numbers, I'll change the whole numbers fractions by just putting him over one right? Nothing wrong there. So if I want to add the's, I know I have to get a common denominator, which I certainly don't have. Here. I have to terms with a common denominator and two other terms with a common denominator. But all four of these don't have a common denominator. If I have denominators of one and x, a common denominator, meaning something that everything can multiply of to would simply be X all right, meaning I don't have to mess with widths of the blue ones. Those are fine. But if I have two X over one and I want to change that into something over X, I would have to take one Times X. But if I'm taking the denominator Times X, I would have to multiply the numerator by the same thing. Two. X Times X would give me two X squared, meaning I need to rewrite both of my lengths as two X squared over X plus two x squared over X. Now I will be able to add those to my widths of one over X plus one over X. Now I can add all these together, right? So working towards adding these all together, Remember, any time we add or subtract fractions, we do not mess with the denominators. We don't combine the denominators. We just rewrite it. We only combined the numerator. All right, so two X squared and two X squared are like terms. Two X squared plus two X squared would be four x squared one and one are like terms. One plus one would be to meaning our perimeter is four x squared plus two over X. That is our perimeter. Get this out of the way. Well, box this off. So now the other thing we need to do is find our area. Remember, that area of a rectangle is found by taking length times width. So we only need to use each of these once, unlike with perimeter. So it's simply going to be our length of two X times are with of one over X. Again, we should probably represent two x as a fraction as two x over one. Remember, when we are multiplying fractions or multiplying rational expressions, you simply multiply. The two numerator is together. Multiply the two denominators together meaning two x times one is two x one times X is X. Looks like I can actually simplify this because thes exes can cancel out. So my final answer for my area would simply be to, and that would be our perimeter and our area.

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