1 00:00:05,609 --> 00:00:07,859 Robert McKeown: Hello, and welcome to ALEKS 2 00:00:07,859 --> 00:00:10,379 walkthrough video number two. I'm Professor 3 00:00:10,379 --> 00:00:12,299 Robert McKeown, and I'm very happy that you 4 00:00:12,299 --> 00:00:15,209 decided to check out my video. Today we're going 5 00:00:15,209 --> 00:00:17,939 to take a closer look at equations. And we're 6 00:00:17,939 --> 00:00:20,909 also going to learn about mathematical notation. 7 00:00:21,329 --> 00:00:24,959 We can think of mathematical notation as a 8 00:00:24,959 --> 00:00:27,779 language, just like English, or Mandarin, or 9 00:00:27,779 --> 00:00:30,359 French is a language. It's the language of 10 00:00:30,359 --> 00:00:33,389 mathematics. And so I hope by the end of today's 11 00:00:33,389 --> 00:00:35,519 video, or perhaps after a little bit of 12 00:00:35,519 --> 00:00:39,629 practice, you'll become comfortable translating 13 00:00:39,659 --> 00:00:42,899 English into math notation, and translating math 14 00:00:42,899 --> 00:00:49,919 notation back into English. Now, I'm not going 15 00:00:49,919 --> 00:00:54,629 to try and cover every single topic. In ALEKS, 16 00:00:54,629 --> 00:00:58,169 I'm just going to cover the topics that I feel 17 00:00:58,169 --> 00:01:03,779 are most helpful to you. Either because they're 18 00:01:03,779 --> 00:01:06,389 important or because they're fundamental. And by 19 00:01:06,389 --> 00:01:10,229 learning them, it's easy to go further into more 20 00:01:10,229 --> 00:01:14,849 detail. As usual, with these videos, what I 21 00:01:14,849 --> 00:01:18,209 expect from you, I expect you to have your 22 00:01:18,209 --> 00:01:22,799 pencil paper, you can print out the slides. If 23 00:01:22,799 --> 00:01:25,049 you have lots of devices, maybe you want to even 24 00:01:25,049 --> 00:01:27,719 write your notes on a second device on a tablet 25 00:01:27,719 --> 00:01:31,229 or something like that. Whatever works for you, 26 00:01:31,289 --> 00:01:33,419 but I highly recommend you work along with the 27 00:01:33,419 --> 00:01:36,059 problems or try the problem first, and then 28 00:01:36,059 --> 00:01:40,649 watch the video. So if you're ready, I'm ready. 29 00:01:42,659 --> 00:01:46,349 Mathematics has its own language of symbols. 30 00:01:46,559 --> 00:01:50,039 These symbols, give you instructions and 31 00:01:50,039 --> 00:01:56,159 describe some of the mathematical values and 32 00:01:56,159 --> 00:02:00,149 variables that you'll be working with. Like any 33 00:02:00,149 --> 00:02:04,229 language, we can translate it into English. If 34 00:02:04,229 --> 00:02:07,919 you look at the slide in front of you, you can 35 00:02:07,919 --> 00:02:11,639 see some mathematical symbols in the left 36 00:02:11,639 --> 00:02:16,919 column, how we would the English expression so 37 00:02:16,919 --> 00:02:20,009 how we would actually say these things, and then 38 00:02:20,009 --> 00:02:22,499 I've given you a little bit of their meaning. 39 00:02:22,949 --> 00:02:26,549 And so I want to take a few moments to explain 40 00:02:26,549 --> 00:02:29,369 to you interval and set notation that'll help 41 00:02:29,369 --> 00:02:34,289 you answer some of the questions and ALEKS take 42 00:02:34,289 --> 00:02:37,019 a moment and familiarize yourself with the 43 00:02:37,019 --> 00:02:45,599 table. Looking at the example, I can write this 44 00:02:45,599 --> 00:02:49,649 mathematical symbol in English, which I'm going 45 00:02:49,649 --> 00:02:57,299 to do for you now this is an interval 46 00:03:02,400 --> 00:03:11,370 and four is the low point or the lowest value 47 00:03:12,360 --> 00:03:21,300 and the interval runs up to its largest value. 48 00:03:24,180 --> 00:03:31,500 And since it really has no upper limit, there is 49 00:03:31,500 --> 00:03:35,700 no largest value. We represent that concept with 50 00:03:35,700 --> 00:03:39,060 the infinity symbol, right. So infinity symbols 51 00:03:39,060 --> 00:03:41,700 are not numbers, infinity symbol represents a 52 00:03:41,700 --> 00:03:54,660 concept. No upper limit on this. 53 00:04:02,340 --> 00:04:04,410 The next thing I want you to notice is that we 54 00:04:04,410 --> 00:04:10,170 have a square bracket. square bracket represents 55 00:04:10,170 --> 00:04:13,170 a close bracket. And so what does this mean? 56 00:04:13,170 --> 00:04:26,430 This means the interval includes the value for 57 00:04:33,570 --> 00:04:35,760 so in English, 58 00:04:41,190 --> 00:04:53,670 x is in. Right That's our n symbol the interval 59 00:04:58,050 --> 00:05:02,970 for To infinity, 60 00:05:08,730 --> 00:05:11,310 where infinity is representing a concept that 61 00:05:11,310 --> 00:05:14,190 there's no, there's no upper limit, whatever 62 00:05:14,190 --> 00:05:17,670 number however big a number, you can imagine, X 63 00:05:17,670 --> 00:05:20,550 can take on that value, there's one more thing 64 00:05:20,550 --> 00:05:24,660 I'd like to show you. And that's an alternative 65 00:05:24,660 --> 00:05:30,240 way of expressing this thing. Mathematically, 66 00:05:30,840 --> 00:05:37,290 and so I could write x is greater than and equal 67 00:05:37,290 --> 00:05:40,620 to, or I should say or equal to x is greater 68 00:05:40,620 --> 00:05:49,890 than or equal to four. And if I write that, you 69 00:05:49,890 --> 00:05:55,170 know, these things are equivalent. So that's an 70 00:05:55,170 --> 00:05:57,180 alternative way of writing it. And that's going 71 00:05:57,180 --> 00:05:59,130 to come in handy when you look at the problems 72 00:05:59,130 --> 00:06:04,230 on ALEKS. And when they ask you to go from an 73 00:06:04,230 --> 00:06:06,660 expression like x is greater than or equal to 74 00:06:06,660 --> 00:06:09,900 four to writing the the set or interval 75 00:06:09,900 --> 00:06:17,070 notation. Now let's go to the next slide. And 76 00:06:17,070 --> 00:06:19,650 let's take a look at the next example. So the 77 00:06:19,650 --> 00:06:23,430 table is the same. But the example down here is 78 00:06:23,430 --> 00:06:28,290 a little bit different. So let's go through this 79 00:06:28,290 --> 00:06:34,200 example. Now, the variable v is in the interval 80 00:06:34,710 --> 00:06:41,400 from negative five to negative two. Now, I said 81 00:06:41,400 --> 00:06:45,630 that, but what about the round brackets and the 82 00:06:45,630 --> 00:06:51,960 square brackets. So now what values can be take 83 00:06:51,960 --> 00:06:59,880 on? Well, we've got around bracket. And so v 84 00:07:01,740 --> 00:07:10,200 must be larger than negative five. And we've got 85 00:07:10,200 --> 00:07:16,920 a square bracket over here. And so v has to be 86 00:07:16,920 --> 00:07:26,460 less than or equal to negative two. Now, here I 87 00:07:26,460 --> 00:07:29,520 am on ALEKS. And so we have a question from 88 00:07:29,520 --> 00:07:33,360 Alex. It says solve the compound inequality. And 89 00:07:33,360 --> 00:07:36,570 so we're given two equations. This time, both 90 00:07:36,570 --> 00:07:41,880 equations have X as the unknown variable. And 91 00:07:41,880 --> 00:07:44,010 then we're told write this solution in interval 92 00:07:44,010 --> 00:07:48,690 notation. If there is no solution to these two 93 00:07:48,690 --> 00:07:54,480 inequalities, enter that funny looking zero, or 94 00:07:54,480 --> 00:07:57,300 that circle with a line through it, which is our 95 00:07:57,300 --> 00:08:00,390 null set, also sometimes called the empty set. 96 00:08:02,070 --> 00:08:04,620 Now you are going to work on this problem on a 97 00:08:04,620 --> 00:08:07,320 piece of paper, I'm going to take us back to our 98 00:08:07,320 --> 00:08:11,850 slides. And so here we are in the slides. And we 99 00:08:11,850 --> 00:08:16,170 want to write the solution to this in interval 100 00:08:16,230 --> 00:08:22,560 notation. So we'll start with this equation on 101 00:08:22,560 --> 00:08:25,710 the left hand side, I've got three X, I'll 102 00:08:25,710 --> 00:08:30,030 subtract six of both sides of the equation. So 103 00:08:30,060 --> 00:08:31,890 I'll just write that as 104 00:08:46,680 --> 00:08:51,540 we have x must be less than six. Now let's take 105 00:08:51,540 --> 00:08:55,050 a look at our other expression. So I see I've 106 00:08:55,050 --> 00:08:57,930 got a two x minus two on the left hand side, I'm 107 00:08:57,930 --> 00:09:00,480 going to add two to both sides of the equation. 108 00:09:00,480 --> 00:09:04,650 So I'm going to have two x is less than or equal 109 00:09:04,650 --> 00:09:11,280 to negative eight plus two. So that gives me two 110 00:09:11,280 --> 00:09:17,940 x is less than or equal to negative six, and x 111 00:09:18,000 --> 00:09:21,330 is less than or equal to divide in both sides of 112 00:09:21,330 --> 00:09:25,800 the equation by two, I get negative three. Now, 113 00:09:25,800 --> 00:09:28,980 the question had a very important instruction, 114 00:09:29,250 --> 00:09:32,160 very important instruction that I have ignored 115 00:09:32,160 --> 00:09:37,830 so far, and that is right here. So the question 116 00:09:37,830 --> 00:09:41,460 is saying that either the question on the left 117 00:09:41,490 --> 00:09:44,400 is correct or true. And the question on the 118 00:09:44,400 --> 00:09:48,360 right is, or the question on the right is 119 00:09:48,360 --> 00:09:54,420 correct. And true. So x must be less than six, 120 00:09:54,450 --> 00:09:58,620 or it must be less than and equal to negative 121 00:09:58,620 --> 00:10:04,440 three. So I'll draw a real number line. And I'm 122 00:10:04,440 --> 00:10:11,160 going to have six over here, and negative three 123 00:10:11,250 --> 00:10:16,410 over there. Now, if I want to start with this 124 00:10:16,410 --> 00:10:22,800 equation, now I'm going to write the equation 125 00:10:24,090 --> 00:10:29,580 over here. And I'll write this one over here as 126 00:10:29,580 --> 00:10:30,030 well. 127 00:10:35,220 --> 00:10:42,090 And so you can see that if x is less than six, 128 00:10:46,170 --> 00:10:50,490 the inequality is satisfied. But I'm going to 129 00:10:50,490 --> 00:10:55,230 draw a circle, an empty circle, here to 130 00:10:55,230 --> 00:10:57,300 represent that this is open. 131 00:11:02,730 --> 00:11:06,000 And that's an analogy. You can think of that as 132 00:11:06,000 --> 00:11:09,090 the an open bracket that we saw previously. 133 00:11:14,520 --> 00:11:20,280 Then over here, x can be equal to negative 134 00:11:20,280 --> 00:11:28,260 three. So I'm going to draw a solid.to represent 135 00:11:28,260 --> 00:11:35,700 it as being closed. So a circle or a dot with 136 00:11:36,150 --> 00:11:39,600 hollow in the middle, that's open, solid dots 137 00:11:39,600 --> 00:11:43,410 going to be closed. Now we can go over to ALEKS 138 00:11:43,440 --> 00:11:46,080 and solve the question or I should say put the 139 00:11:46,080 --> 00:11:52,350 answer, ALEKS. Before we move over to ALEKS, 140 00:11:52,350 --> 00:11:54,420 though, maybe I should point out to you that, 141 00:11:54,510 --> 00:12:00,120 notice that if x is less than or equal to three, 142 00:12:01,170 --> 00:12:18,330 then x is also less than six. And since either 143 00:12:18,330 --> 00:12:22,980 one of these inequalities has to hold, the 144 00:12:22,980 --> 00:12:27,780 correct answer is going to be x must be less 145 00:12:27,780 --> 00:12:41,040 than six. And so x is in the set negative 146 00:12:41,340 --> 00:12:48,030 infinity to six with a round bracket to 147 00:12:48,030 --> 00:12:52,290 represent that x cannot take on the value of 148 00:12:52,290 --> 00:12:57,030 six, it's an open set can't take on the value 149 00:12:57,030 --> 00:13:04,410 six. So here we are in ALEKS, we have the same 150 00:13:04,410 --> 00:13:09,930 question that we're given before. What is the 151 00:13:09,930 --> 00:13:13,740 interval here that we're looking for? Well, we 152 00:13:13,740 --> 00:13:21,720 know that it's going to be rounded brackets. and 153 00:13:23,430 --> 00:13:28,830 the value of x can be as low a number as you can 154 00:13:28,830 --> 00:13:33,150 imagine, so I'm going to put a negative infinity 155 00:13:33,150 --> 00:13:39,960 symbol in there. And the maximum value it can 156 00:13:39,960 --> 00:13:43,230 take on is something just a little bit less than 157 00:13:43,230 --> 00:13:45,510 six. And so we represent or I should say, a 158 00:13:45,510 --> 00:13:49,350 little bit less than Oh, no, sorry, that was 159 00:13:49,350 --> 00:13:55,500 six. any value less than six. Now I'll check my 160 00:13:55,500 --> 00:14:00,990 answer. And we got the correct answer on ALEKS. 161 00:14:01,350 --> 00:14:04,410 The interval of x is going to be between 162 00:14:04,410 --> 00:14:09,210 negative infinity and six but it cannot take on 163 00:14:09,210 --> 00:14:11,190 the value six. So we have the rounded bracket.