WEBVTT 00:00:02.620 --> 00:00:06.350 In the figure, the perimeter of the rectangle is less than that of the triangle. 00:00:07.400 --> 00:00:12.160 Write an inequality that can be used to find the range of values that 𝑥 can take. 00:00:12.770 --> 00:00:14.800 Then, solve your inequality. 00:00:16.370 --> 00:00:21.350 So the key here is that we’re told that the perimeter of the rectangle is less than that of the triangle. 00:00:22.580 --> 00:00:28.310 So before we can set up our inequality, what we need to do is work out the perimeter of the triangle and the rectangle. 00:00:28.970 --> 00:00:32.320 Well, the perimeter of the triangle is the distance around the outside. 00:00:32.350 --> 00:00:34.970 So what we do is we add together the side lengths. 00:00:36.350 --> 00:00:41.140 So we have 𝑥 plus one plus 𝑥 plus two plus 𝑥 plus two. 00:00:42.750 --> 00:00:46.880 So then, if we collect all the 𝑥 terms, we’ll have 𝑥 plus 𝑥 plus 𝑥, which is three 𝑥. 00:00:48.180 --> 00:00:50.470 And then, we collect our numeric terms. 00:00:50.470 --> 00:00:53.210 So we have one plus two plus two, which is five. 00:00:53.570 --> 00:00:57.870 So therefore, the expression for the perimeter of the triangle is three 𝑥 plus five. 00:00:59.500 --> 00:01:04.070 Well, now, if we move on to the rectangle, the first thing we do is we have to add on the other dimensions. 00:01:04.440 --> 00:01:09.100 Because it’s a rectangle, we know that the opposite sides are equal in length and parallel. 00:01:09.580 --> 00:01:13.550 So we’ve got 𝑥 minus one, 𝑥 minus one, 𝑥 plus two, and 𝑥 plus two. 00:01:15.110 --> 00:01:25.580 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. 00:01:25.820 --> 00:01:27.100 And that’s cause we got four 𝑥s. 00:01:27.100 --> 00:01:30.420 Then, we’ve got two plus two, which is four, take away one is three. 00:01:30.420 --> 00:01:32.260 Take away another one is two. 00:01:32.480 --> 00:01:35.830 Okay, so now, we have our expressions for both of our perimeters. 00:01:36.100 --> 00:01:37.480 Let’s form our inequality. 00:01:39.390 --> 00:01:45.010 Well, our inequality is gonna be four 𝑥 plus two is less than three 𝑥 plus five. 00:01:45.490 --> 00:01:48.020 And that’s because four 𝑥 plus two is perimeter of the rectangle. 00:01:48.020 --> 00:01:51.380 And we’re told that the perimeter of the rectangle is less than that of the triangle. 00:01:52.880 --> 00:01:57.870 It’s worth noting that the notation we use for inequality did not have a line underneath. 00:01:57.970 --> 00:01:59.970 And that’s because it’s less than. 00:02:00.190 --> 00:02:02.920 If it has a line underneath, then it is less than or equal to. 00:02:03.200 --> 00:02:05.340 Okay, great, we’ve solved the first part. 00:02:05.470 --> 00:02:07.880 Now, what we need to do is solve the inequality. 00:02:09.800 --> 00:02:14.070 Well, to solve the inequality, first of all, we look to see where the most 𝑥s are. 00:02:14.710 --> 00:02:17.730 And we can see that the most 𝑥s are on the left-hand side. 00:02:18.280 --> 00:02:23.770 In that case, what we’re going to do is subtract three 𝑥 from both sides of our inequality first. 00:02:25.060 --> 00:02:28.080 And when we do that, we get 𝑥 plus two is less than five. 00:02:28.890 --> 00:02:31.080 So great, now, what’s the next step? 00:02:31.110 --> 00:02:35.890 Well, the next step is to subtract two from each side of the inequality because we want the 𝑥 on its own. 00:02:37.650 --> 00:02:40.300 And when we do that, we get 𝑥 is less than three. 00:02:42.020 --> 00:02:44.410 So therefore, we’ve solved the second part of the question. 00:02:44.830 --> 00:02:54.140 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. 00:02:54.600 --> 00:02:58.830 And the range of values that 𝑥 can take are 𝑥 is less than three.