1 1 Real Numbers Algebra Essentials

0:00 welcome everyone uh this is Professor Ridge and today's lesson is on real numbers the algebra Essentials okay so let's start with the first learning objective classifying a real number as a natural whole integer rational or irrational number okay the first set of numbers that we'll talk about is What's called the natural numbers and and if you think about it when uh in human development the first use of numbers was probably to go out and count things so we call those the natural numbers or counting numbers and they're simply the set that

0:48 we of numbers that we use for counting so the natural numbers are 1 2 3 4 5 and so on then uh the next set of numbers is whole numbers now believe it or not this took many many years to come up with a symbol to represent nothing so the only difference between the natural numbers and the whole numbers is the number zero that represents nothing so 1 2 3 4 five so

1:34 on next we wanted to introduce numbers to represent negatives right oing and so if we add the negative numbers of these that or we call them the Opposites of the whole numbers so the opposite whole numbers um 1 --2g -3 so on that set gives us the integer so the integers are the negative of the whole number so -5 -4 -3 -21 0 plus the whole numbers so

2:21 basically we're adding just the negatives to this whole number set one 2 3 four five so on okay next we're going to add the numbers in between not all of them but um for instance if I wanted to have a portion of one 1/2 1/3 or one and a half two 2.7 those numbers we call the rational numbers so the rational numbers include include all the integers plus those fractions in that are in

3:09 between now as far as notation goes um for these sets this is what we call roster notation by the way um where we just explicitly list every member of the set um so for um rational numbers you can't list them all so we have to describe it and this notation we're using will be called roster I mean set builder notation so you're going to describe what does it mean to be a member of the set so it' be a rational number it's basically a fraction so you got M over n it's a ratio that's where rational comes from um and then M and this straight bar means such that so this is the set of all M over n such

3:58 that m and N so now we're describing uh what M and N are so M and N are ra or integers and uh we can't divide by zero so we want to make sure n's not zero okay so that describ describes the set of rational numbers so again some examples for rational numbers would be like 1 over two 1/2 so you see that fraction um and again all the integers

4:43 are part of rational numbers because you can take every uh R every integer like seven and write it as a ratio of two integers so that's a rational number so it includes all the integers same thing for like -3 you can write -3 is -3 over 1 so you can write any integer as a fraction by putting it over one so all these are all the integers are also rational numbers now the next set is called the irrational numbers irrational numbers

5:29 now this one is basically simply set of all H's is the way this book uses where H is not a rational number I should add it but it has to be a real number it's a real number that's not rational and we'll get more into that later so what is a rational number or irrational number sorry um basically it was kind of discovered if you had a square so the Greeks did a lot of geometry I believe they're the first one that noted maybe maybe not um so if this

6:17 side was one one and they figured out this number you couldn't represent the length of this diagonal by um anything that was a um basically a ratio no matter how you scaled it you couldn't make this a ratio of two integers all right so this is actually square otk of two that length and square otk of two is what we call um an irrational number and it is if you put it in decimal form I should say about equal to 1.414 and it goes on forever and ever and ever it's a non-terminating this something you might want to

7:02 real all right let's look at example three from the text um differentiating rational and irrational numbers determine whether each following number is rational or irrational if it's rational determine whether it's terminated or repeating decimal now um probably the easiest way right if you just got a calculator um Square < TK of 25 is just five it's a whole number so this is equal to five so you can see that it it's a terminating decimal if you think of it as a decimal number it stops it's just five okay so that's a term that's

7:48 rational and it's terminates it's a terminating uh decimal number okay 39 over 9 if you punch that in your calculator it will be here let me get a calculator open here so Desmos is my uh one of my favorite graphing calculator programs um you just go to desmo do desmos.com and um it's free there's no charge for it so I actually use this all the time in the classroom all right so if I wanted to do 33 ided by 9 Ian you can see that's not going to be a

8:33 whole number and then you see it's 3. 6666 and it's going to repeat forever here the calculator's rounding it but it's 3.66 so that would be a um repeating decimal was it 3. 6666 put a line over the six right um so again it be rational so to be irrational needs to be non-terminating non- repeating so it has to go on forever and there can't be a pattern in it if we did square root of 11 you can see in decimos here let's do square root you just type there's a tool bar at the bottom you hit square

9:21 root or you can type sqrt uh square otk of 11 3316 62479 036 and it'll just keep going forever and ever and ever so that one is irrational and it's non terminating okay 17 over 34 so one thing to know too if it's written as a ratio of two integers you know that's the definition of a rational number so it's obviously going to be either a terminating decimal or a repeating decimal right so 17 over

10:09 34 where did my decimals uh calculator go hang on all right so if we put in oh we don't want square root so we want what is it 17 over you just have to divided by um 34 and oh that is 0.5 so that is I'm laughing at myself I should have recognized that I'm usually pretty good with halves right that's one half so 0. five so obviously that terminates and then um this one it's 3 330

10:56 3330 so even though that there's a pattern in there it's not technically repeating so that one we would consider that one we would consider um irrational okay so that is irrational so when we say repeating we mean like um two digits repeat forever or three digits repeat forever but this is um you can't say three repeats cuz then it would be 3 3 3 3 you can't say 303 repeats cuz that would be right to there but then that gets messed up you see so um that would not be considered a

11:43 repeating a decimal all right let's move on real numbers so the sets of rational and irrational numbers together make up the set of real numbers so we can represent every real number on what we call the real number line and it classify we can break it into three subsets we can have zero all by itself and then everything to the right of zero would be the positive real numbers and everything to the left of zero would be the negative real numbers so zero is the what we call the origin or if it's zero right it's just it's

12:29 neutral it's not positive or negative everything to the right is positive everything to the left is negative so if we were to um again every point on this represents a real number so for instance this might be right here might be 0. five oops let me get in my so here 0 five or one half same thing so that point is 0 five um the square root of two is about 1.4 so roughly there just kind of eyeballing it so which is about 1.4 um if you wanted to do 9 Hales so N9

13:16 9 ID 2 is 4.5 so you can think of right there would be nine halves so those are all positives if I did a negative say1 13 be right about there okay and then if I did um Square t negative square < TK of five hang on okay that's about 2.2 36 so negative so right roughly there so again every point every real number can be represented by a point on the real number line anything to the

14:02 right of zero is positive anything to the left of zero is negative zero is neutral doesn't have a sign it's neither positive nor negative all right let's look at example four classifying real numbers classify it's either positive or negative and either as rational or irrational so again this um the minus sign you can interpret it as also negative so - 10/3 that's a rational it's a ratio of two integers so that would be let's see that's either positive or negative and rational or irrational so that would be a negative rational

14:49 number okay so negative and rational oops negative that's horrible negative in rational okay square root of five well we just talked about that it's 2.23 six it goes on forever so that's irrational that's positive right there's no negative sign so that'll be positive and irrational oh and then does it lie to

15:35 the left or right of zero so if it's negative it's to the left so this would be this one to the left and this one to the right um 289 now you got to be careful just because you have the square root right um if you know your squares you this should be kind of say hey that's a perfect square so if I were to punch it into a calculator you're going to get a whole number which is rational so let's see here so if I do square root of 289 that's 17 right and then the negative in front would just make it - 17 so that one is okay I'm not going to

16:20 write it just say it so this one will be negative and irrational no rational maybe I should write it so it's negative and rational and then it'll be to the left of zero on the number line Pi all right the next one here Pi is um represents the number Pi represents the number roughly 3.14159 so on okay and it is an irrational number and then if you multiply that by six it'll still be an irrational number so in fact let's let me show you on here so if

17:07 we do Nega 6 times times pi you can see this is going to be a non-re repeating non-terminating decimal number so that one is irrational and it's negative so be to the left and then the last one do we have any pattern in here no there's no repeating digits and it's dot dot dot so it goes on forever so that one will be irrational and it's positive and it'll be to the

17:52 right all right I always like to look at this slide here um so if we think of the real numbers is the the subsets of the real numbers we had we started out with the natural numbers that's this inside one here so that was the set of numbers that we use for counting and we when you start counting you start with one not zero so one 2 3 then we talked about all right let's add in zero to the set we called that the whole numbers so the only thing in this set is uh extra mean is zero right so it includes all the naturals plus zero that makes the whole numbers then we added all the negative numbers that gets us

18:39 the irrationals and we're using I for irrational W for whole in for natural and then the set Q um is the set of rational numbers so that's all the fractions that be can that you can write as a ratio of two integers okay and then everything outside of outside of that are so basically this circle everything inside this the biggest circle there or oval I should say is uh rational numbers so that includes all the um integers all the whole numbers and all the um natural numbers did I say I was

19:26 irrational earlier it's I is for the the integers okay some books use different letters um anyway doesn't matter so um anything that's outside the biggest circle so inside the biggest circle are all the rational numbers and then outside would be the irrationals and together everything in here plus everything outside would be the real numbers so the the universe in this case would be the real numbers um later in the text we'll talk about imaginary so you got the real and the imaginary but for right now just talking about real numbers and so um this is kind of how it splits up so again a a real number can

20:14 either be one of two things it could be ra it could be rational or irrational can't be both but as far as rationals you could be a natural number a whole number an integer right those are all included in rational numbers so for instance one is a natural number it's also a whole number it's also an integer and it's also a rational okay so that's these are considered subsets when they go inside like that okay let's move on to this next objective uh perform calculation calculations using order of operations

20:59 the order operations or operations in mathematical Expressions must be eval evaluated in a systematic order which can be simplified using the acronym pimus NOW be careful with this um because when hear lots of uh people talk about oh pimas or you can use um please excuse my dear ant Sally the problem is there's basically four levels of precedent for doing order operations multiply and divide are considered the same priority level right so you don't necessarily multiply first then divide so the tie breaker I wish they wrote it in the book here the tiebreaker is from

21:46 left to right and then the same thing for addition and subtraction and we we do a couple examples I'll show you what I mean but it's really important that you remember it's multiply and divide are on the same level right so think of it as four levels you're going to do everything in parentheses now parentheses you can actually broaden that term and it's grouping symbols so it could be parentheses brackets the fraction bar is a grouping symbol okay then you do all your exponents and then that's the second level then the third level you do all the mulation or division from left to right and then addition and subtraction is your fourth level so that's the last and

22:34 again from left to right okay so the next little box in the textbook um did clarify this a little bit so again so again think of it is four levels right first you're going to do any thing within grouping symbols so that's the parentheses um let's see let's write down so it could be parentheses it could be bra ETS usually use like these for nesting like um this if you had parentheses within parentheses you would use brackets just to help distinguish it um braces right absolute value um I'm not to write that down um the division bar fraction bar it

23:22 separates into a numerator and a denominator so it groups it into those two things then your second step simplify expressions containing exponents or radicals so radicals we'll talk about that later too radicals can be thought of as a exponent as a fraction exponent or a rational exponent and then you're going to perform any multiplication division from left to right as you can see right there and then once that's done you're going to go to the next step and do add add and subtract from left to right okay okay so let's look at example six from the textbook uh

24:08 a we want to use op or operations to evaluate the expression so if we look at this so again we're going to start we're going to start with uh grouping symbols so we got to do inside the grouping symbol okay so inside grouping symbols this something else we should mention you follow or operations so if there here's a grouping symbol if there's another grouping symbol inside you're going to do inside there first so let me show okay let me just show you real quick what I'm talking about let's say we had something like five plus uh 2 um * 3 + 4 in parentheses like that so

24:54 this is a grouping symbol and then we're going to do inside this grouping symbol first so we're going to do that first get an answer okay and then we're going to work work from there so that will get us 5 + 2 * 7 so you can replace if you have a number next to a parenthese you can replace that with a times once you make this one number okay now in here we're going to still follow or order operations so there's no more grouping symbols there's no exponents so then we're going to do multiply and divide from left to right so that's a plus sign not a divide so 2

25:39 * 7 is 14 so 14 and then uh 5 + 14 would be 19 and then once that's one number you can drop the parentheses now if there's a number right in front of it you could put a t you would put a time so if this was like three if I had a three out here then it' be 3 * 19 all right so inside this parentheses we do 3 * 2 is 6 and then inside this parenthese we do 6 + 2 which is

26:27 8 and you can leave this is with the parentheses or I could drop the parentheses and put a dot there for times or I can just end my head think four times eight all right so we're done with grouping symbols now so when I say grouping symbols meaning there has to be more than a single number in it right so if I have a single number in it then you can kind of basically you can drop the the grouping symbol unless you have other things like uh you got to be careful with the the exponents so this is 6 sared 6 * 6 is 36 and then 4 * 8 is 32 oh I'm skipping a

27:12 step okay so now so now we did the exponents okay then we're going to perform any multiplication and division from left to right so I'm going to do this one a little experience you'll know that yeah I could have done it then and then 36 minus uh 32 is 4 and that's our final answer all right let's look at another one okay so we'll look at C this is example six C in the book um so we got grouping symbols remember I said the absolute value grouping symbol so we need to come we need to take care of that first so 6 minus the absolute value

27:59 now 5 - 8 if you forgot your rules right um subtract means add the opposite so I'm thinking of this as 5 plus A8 signs are different so I subtract 8 - 5 is three and I take the sign of the larger absolute value number so Eight's bigger than five so I take the sign of 8 so that' be -3 and then four okay so we're still UND doing grouping so that grouping symbol's done now here's another grouping symbol so I got to do this one 4 minus 1 is three okay now here's you got to be careful

28:45 here the absolute value of -3 is just positive3 or just three you can think of it as that and then um remember I made this one number so I can drop the parentheses but if there's something attached to it in front there I got to put a times so it's 3 * three all right so grouping symbols are done now step two we're going to do anything that has exponents there's no exponents then we go to step three multiply and divide from left to right so here's a times so 3 * 3 is

29:31 nine okay so that step's done now add subtract from left to right so we get 6 - 3 is 3 + 9 is 12 excuse me okay so that's my final answer there I guess I could be putting equals over here and I like you to box your answer or Circle it when you um get complete a problem okay okay so example 6D so now notice this has a big fraction bar here so that's a grouping symbol so I got to treat the top the numerator the bottom

30:18 the denominator think of them as two separate problems so that's in my first step so on the top so basically if I uh write this out without the um fraction bar it's basically the top in parentheses divided by the bottom in parentheses that's the way you write it okay so in let's look at the numerator so you can kind of think of it as two separate problems here so let's see 14 okay so there's in the numerator there's no grouping symbols there's no radicals so I'm going to do multiply Divide from left to right so 3 * 2 is 16 I mean sorry six okay and then uh that's done so now

31:08 we're going to do subtract from left to right so 14 minus 6 is what is that um eight let me8 okay now on the bottom again so we're grouping symbols so we're doing the bottom group um inside that grouping symbol we got to do order operations so let's see exponents so 3^ SAR is 9 we leave everything else the same and then okay exponents are done now we're going to do multiply divide left to right so 2 * 5 is 10 so now we

31:55 got 10 - 9 okay then we got to do add subtract from left to right so eight over leave everything else the same 10 - 9 is 1 okay now so basically we did we did the numerator we did the denominator now we're going to divide these two numbers so 8 ID by one which is just simply eight that's our final answer all right so let's look at e now all right so we got a grouping symbol here around the five * three a bracket another grouping symbol and then nested inside that one

32:43 there's this 6 - 3 so um we can kind of think of them as here's another way to think of it it's not written in there you could think of whenever you see this m you could think this is three D separate problems that one that one and that one okay so for the first set I got to do inside the grouping symbol 5 * 3 is 15 and then now since that's one number I can drop the parentheses and put times 15 now in this grouping symbol we got minus 2 inside here I got to do 6 Min - 3 is three and I could drop that parentheses

33:31 and this one there's nothing to do so just write it down okay now we're still working grouping symbols so I got to make this take care of this so in here I did the grouping symbol now I'm going to do exponents so I got to do four squar first again if you want you can go ahead and do 7 * 15 and make that one number but we'll we'll come back to that and then um 3 - 4^ 2 16 and then um 3 - 16 is -13 so again it's 3 + -16

34:23 you got a positive and a negative subtract 16 - 3 is 13 and you take the sign of the larger which would be negative which would be 16 so it be -3 then plus one all right so now we're going to okay so our next step is Let's see we did all the grouping symbols so this is one number I can drop it and put times but since this is negative I'm going to go um I'm just going to leave it the way I have it written okay so we did all the

35:10 parentheses there's no more exponents so now we're going to do multiply and divide from left to right so 7 * 15 is um what is that 105 minus 2 * 13 is -26 or you can think of this is the way that I think of it I think of it as2 see this is I would think of it as -2 * ne3 right I don't rewrite this but in my head so negative and negative make positive and then 26 + 1 now I'm going to add subtract

35:57 ract from left to right so 105 + 26 + 1 and that equals 12132 that's my final answer there okay moving on to the next objective use the following properties of real numbers communative associative distributive inverse and identity the communative property of addition stat that numbers may be added in any order without affecting the sum so we write that in notation as a + b is equal to B+ a so we we should have had some qualify A and B are any real

36:43 numbers so these are properties of real numbers okay so an example would be um like three + 5 is the same thing as 5 + 3 we know that 3 + 5 is is 8 5 + 3 is also 8 okay so you can see that that doesn't the order that you're adding the numbers doesn't affect the outcome okay so that's the first one and this also works with multiplication so multiplication is also commutative so if I had like um 7 *

37:28 three that's the same thing as 3 * 7 so the order doesn't matter 21 you get 21 either way okay and again it's important to note that neither subtraction nor division is communative so for example if I did all right let's say um 6 minus 6 - 2 and then you could see that if I did 2 - 6 you could see 6 - 2 is 4 but 2 - 6 is -4 so those are not equal okay and same thing with M uh Divi dividing if I did 8 ided

38:16 4 I get two and if I did four divided by 8 I get 0.5 or 1/ 12 so those are not equal 8 / 4 is not equal to 4 / 8 okay so it's not communative all right the sociative property of multiplication tells us that it doesn't matter how we group numbers when multiplying we can move the grouping symbols to make calculation easier and the product Remains the Same so let's look at an example let's say I did something like uh 2 *

39:05 3 times 4 so M you know our four basic math operation add subtract multiply divide they're actually what we call binary operations you need two numbers so 2 * 3 for instance will get a six right so we don't memorize multiplication tables 2 * 3 * 4 and know that answer right we got to stop and we got to just do two at a time so the associative property says if it's all multiplying right so property multiplication it doesn't matter how you group those two at a time you could do the first two 2 * 3 is 6 and then 6 * 4 is

39:52 24 or I could do it the other way around if I had two * 3 * 4 I can group the three and four together first 3 * 4 is 12 and then times the two is 24 and you get the same answer I wrote These backwards from the way it's up here so you can move that group around so that's the associative property of multiplication and it works with um addition as well okay so if we try something like we'll use same number numbers 2 3 and 4 so if I did 2 + 3 + 4 again binary operations I can only technically do two numbers at a time so

40:40 if I do 2 + three here we'll do it we'll do it um using this property here so 2 + 3 + 4 so if I do the 2 + three first I get five and then 5 + 4 I get nine and then let's look at it this way so if we group the last two 3 + 4 is 7 and then 2 + 7 is nine and you see you get the same number so that's the associative property it doesn't you can rearrange the group and it'll still get you the same sum provided it's all sum so just like with the community property

41:27 it works for addition and multiplication but not for division or subtraction okay so associative property doesn't work for subtraction or division the division property states that the product of a factor times a sum is the sum of the factor times each term in the sum all right so this is a pretty big one so so let's look at numbers here so if I had like 3 * 4 + 5 we're saying that we can say we could take three times each of the addins in that sum so 3 * 4 and 3 *

42:15 5 so if we did this using order operations 4 + 5 is 9 and then 3 * 9 is 27 and let's see if that gets us the same result so using the distributive property we take the three and we're going to multiply it with each distributed so 3 * 4 plus plus 3 * 5 so 3 * 4 is 12 3 * 5 is 15 so 12 + 15 is 27 and you can see you get the same answer no either way right so you can do it using in orderorder operations or we can apply the distributive property and

43:01 we can take that number that's being multiplied on the outside and multiply it with each of the individual U add ends and then add them together okay the identity property of addition states that there's a unique number called the additive identity which is zero that when added to the number results in the original number so the way we write this identity property is any so a can be any number any number plus 0 is equal to the original number right um we should also write it the other way around um we know that um we

43:47 know that addition is communative so 0 plus any number is also equal to itself okay and the identity property multiplication so identity for multiplying is one and then any number times one is itself or again because its multiplication is commutative we could do it the other way around one time any number is the same thing as any number time one which is just itself okay so example if we did like 5 + 0 that's the same thing as zero so this kind of important concept as we get into um algebra we can say x is our variable

44:35 and then if we had x + 0 we know that that's just X we're going to use that all the time and same thing with multiplying if I did 1 * 7 that's seven if I did um 8 * 1 that's 8 and if I'm doing variables 1 * X is just X or x * 1 is just X I should use why how about that so if I did y times 1 is just

45:20 y all right so that's the identity property so the identity for adding is zero the identity for multiplying is one you should know that the inverse property addition states that for every real number a there's a unique number called the additive inverse or opposite so the opposite is the additive inverse those are the same thing okay ad inverse is actually more broad of a term but if we're talking about real numbers the additive inverse is the same thing as the opposite now if we're talking about other things that have additive inverse might be something else okay so anyway back to real numbers um the additive inverse we're going to write it as

46:07 negative a or you should read this as the opposite of a right because it's the additive inverse so it's the opposite of a so if you take any number and add it to its additive inverse to its opposite you're going to get zero so for instance if I took five the additive inverse of five or the opposite of five is neg five and 5 +5 is equal to zero if I did um 7 or let's do -7 the opposite of -7 is the same thing as 7 so that's the

46:53 additive inverse so -7 + 7 is zero the inverse property multiplication holds for all real numbers except except zero because the reciprocal of zero is not defined so you got to know that so zero doesn't have a reciprocal when you're multiplying um and it states for every real number a there's a unique number called the multiplicative inverse or simply the reciprocal again multip IC ative inverse is a broader term when we're talking about real numbers the multiplicative inverse is the reciprocal and we denote it as one over itself one over the

47:40 number and if you take the original number multiply it with its reciprocal you're going to get one which is called the multiplicative identity so if we took 7 times the reciprocal of s which is 17th those would cancel and you get one and if the number was a fraction you do one over 23 which again remember this stands for reciprocal so the reciprocal of 2/3 would be three halves you see you get 6 over 6 and 6 ided 6 is just one okay so that's um oh and then as far as signs what if I did like -5 now be careful the reciprocal

48:27 is 1 over -5 and when you're multiplying negative * a negative makes positive and then the fives cancel and you get one okay so same thing like -58 the reciprocal would be 85s so you get 40 over 40 and negative negative make positive and then you get one here's a nice summary of all the prop properties real numbers we just went over so we got communative property right order doesn't matter so that works for addition multiplication associal property we can change the grouping again addition

49:12 multiplication notice there's no subtraction or division here distributed property the identity property and the inverse property okay moving on to the next um evalu algebraic expressions and I know the lecture is going long but it's recorded so you can pause take breaks you know it's thinking about breaking this up but figure ah you can pause it take a break come back to where you left off and so I'll try to make them a little bit shorter but anyway I apologize for that all right I want to highlight some things from the book here so uh the mathematic expression we've seen have involved real numbers only in

49:58 mathematics we see expressions like x + 5 4/3 pi R cubed or the square < TK of 2 m 3r * n^ 2 in the expression x + 5 five is called the constant or a constant okay so that's a big idea there constant it doesn't vary right so we say it's constant stays the same is just a number it has the same value no matter what x now you got to be careful sometimes we have letters that represent a constant for instance Pi that's a constant even though it has a you know a symbol there um but if it's a if it's

50:45 representing a number that can change or vary we call that a variable okay variable so X is a variable okay an algebraic okay so here's the definition algebraic expression it's just a collection of constants and variables joined together by algebraic operations of you know your math stuff right add subtract multiply divide and of course you can have powers and things like that right okay list the constants and variables for each algebraic expression so again our constants will

51:30 be um we'll make a little table here constants so that' be five for this one it be 4/3 and Pi remember I said Pi is just represents a number that doesn't change 3.14 so on it's irrational um and then in this expression square roots not a number or anything so two is a constant that's it and then the variables all right variables I'm just write be X and then in this one the only variables are and then this three is not a constant it's just it's a power right

52:18 it's a power for the variabl so just ignore it and same thing with the square root it's just it's just a math operation it's kind of like a power actually um so we got m and n as our variables here M and N okay example nine eval evaluate an algebra expression at different values so evaluate the expression 2x - 7 for each value of x so I added this here um to evaluate or evaluate an expression that just means evaluate simply means get a value for it figure out what it's what the value is right so to do it you're just going to substitute the

53:04 given values for the variables right if there's more than one you put them in for all of them and calculate using order operations all right so let's try a couple of these so for a if x is 1 2x - 7 okay so if x is one oh I'm sorry X is zero you just plug in zero for the X and then you work it out to or operations so we do in the parentheses there's no there's nothing to do there so um exponents no exponents multiply so that's zero and then subtract 0 - 7 it's

53:53 just7 all right let's try B 2x - 7 I'm gonna write 4X = 1 so we take the X Out put in one now get in this habit of putting a parentheses when you substitute um if there's negative numbers and things it can help it can keep you from making a simple mistake with signs so this is a good habit to get into just put a blank parenthese for the pr uh the variable and then substitute put in plug in the number that's given for that variable okay or operations again there's um again I'm not thinking this is a grouping symbol now since it's just one number right there's nothing to do so we

54:39 can skip that there's no exponents multiply 2 * 1 is 2 2 - 7 is -5 oh yeah so I'd like you to box your answers all right let's try one more of let's do D I guess I could have done both of them so 2 x - 7 for X is -4 so 2 * -4 this is what I'm saying if you do parentheses right you're not going to confuse this as 2 minus 4 it's 2 * -4 so that's 8

55:27 and8 plus negative so the again in my head without rewriting this I'm thinking as8 plus -7 if both signs are negative you add the numbers the absolute value of the numbers so 8 + 7 is 15 and then you keep the sign so it'll be -5 all right so there's some math 85 review for you okay and on to our last learning objective for today's lesson simplify algebraic expressions so sometimes we can simplify an algebraic expression to make it easier or to to evaluate make it sorry make it easier to evaluate or to use in

56:13 some other way to do so we use the properties of real numbers we can use the same properties and formulas because they contain algebraic expressions so basically I think of simplifying is just how do you can you write it more simply right is there another way so for instance if it's just whole numbers 3 + 5 that's an expression right um but you can write it more simply by figuring out 3 + 5 that's the same thing as eight right so that's simplifying so eight is more simple than 3 + 5 now you do that with algebraic expressions with variables and all that so let's look at some examples there

56:58 okay let's start with a so 3x - 2 y + x - 3 y - 7 okay so in a beginning algebra class right we we have all these properties that we just talked about the um associative and communative and all that and the distributive properties so what we could do is we what we're going to think of this as normally you don't write this down but we want to think of everything as addition because then we have all the communative properties so normally we don't write this but in our head we're thinking

57:44 this okay so everything's in terms of additions so I can rearrange the order put my x's together put my y's together I'm going to leave those in parentheses since they're negatives now um we have associative properties so we can group things so I can look at these and I can I can do the distributive property in reverse and pull out a three oops I don't know what I just did um oh no sorry I'm not going to pull out the three I'm going to pull out the X I'm going to distribute the

58:34 X so oops okay so I'm going to factor out the X now now because of the identity property we can think of this as 1 time x so I'm going to factor out the X which is the opposite it's we're doing the um communative property in reverse and same thing for here going to factor out the Y so I got -2 + -3 and again we don't show all these steps and then 3 + 1 is four so that's how you combine like terms as you're really doing the distributive property

59:19 -2 and -3 is-5 and then again multiplications uh communative so I can switch the order right and it's easier to write instead of writing plus neg5 it's easier to write minus five same thing so this is that same expression more simple so I simplified it so I was just showing all the you know all the properties that we just went over in there so typically when you simplify this the way you would write it you go basically 3x + x is 4x - 2 y + -3 Y is - 5 Y and then my 7 you don't have

1:00:06 to show any of that work all right let's try B now so let's see I'm going to do 2 r - 5 * 3 - R plus 4 I'm not going to show all the detail like there okay my first step is I can distribute that um I can distribute I'm going to think of this as plus and5 going go ahead and write it and I'm going to distribute ne5 so5 * 3 is -15 -5 * R is positive 5r then I got the +4 I'm going to combine my like terms 2 r + 5 R is 7

1:00:56 R and then -15 + pos4 is -11 and then that's my answer okay I think that's enough so that will conclude the lesson for the first one um as far as uh just read the syllabus make sure you know what to do for the homework assignment and I will will see you soon