In the figure, the perimeter of the rectangle is less than that of the triangle. Write an inequality that can be used to find the range of values that 𝑥 can take. Then, solve your inequality.
So the key here is that we’re told that the perimeter of the rectangle is less than that of the triangle. So before we can set up our inequality, what we need to do is work out the perimeter of the triangle and the rectangle. Well, the perimeter of the triangle is the distance around the outside. So what we do is we add together the side lengths. So we have 𝑥 plus one plus 𝑥 plus two plus 𝑥 plus two. So then, if we collect all the 𝑥 terms, we’ll have 𝑥 plus 𝑥 plus 𝑥, which is three 𝑥. And then, we collect our numeric terms. So we have one plus two plus two, which is five. So therefore, the expression for the perimeter of the triangle is three 𝑥 plus five.
Well, now, if we move on to the rectangle, the first thing we do is we have to add on the other dimensions. Because it’s a rectangle, we know that the opposite sides are equal in length and parallel. So we’ve got 𝑥 minus one, 𝑥 minus one, 𝑥 plus two, and 𝑥 plus two. So therefore, the perimeter of the rectangle is gonna be equal to 𝑥 plus two plus 𝑥 plus two plus 𝑥 minus one plus 𝑥 minus one, which gives us four 𝑥 plus two. And that’s cause we got four 𝑥s. Then, we’ve got two plus two, which is four, take away one is three. Take away another one is two.
Okay, so now, we have our expressions for both of our perimeters. Let’s form our inequality. Well, our inequality is gonna be four 𝑥 plus two is less than three 𝑥 plus five. And that’s because four 𝑥 plus two is perimeter of the rectangle. And we’re told that the perimeter of the rectangle is less than that of the triangle. It’s worth noting that the notation we use for inequality did not have a line underneath. And that’s because it’s less than. If it has a line underneath, then it is less than or equal to. Okay, great, we’ve solved the first part. Now, what we need to do is solve the inequality.
Well, to solve the inequality, first of all, we look to see where the most 𝑥s are. And we can see that the most 𝑥s are on the left-hand side. In that case, what we’re going to do is subtract three 𝑥 from both sides of our inequality first. And when we do that, we get 𝑥 plus two is less than five. So great, now, what’s the next step? Well, the next step is to subtract two from each side of the inequality because we want the 𝑥 on its own. And when we do that, we get 𝑥 is less than three. So therefore, we’ve solved the second part of the question.
So we can say that the inequality that’s used to find the range of values that 𝑥 can take is four 𝑥 plus two is less than three 𝑥 plus five. And the range of values that 𝑥 can take are 𝑥 is less than three.