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>> Hello there. My name is Shannon Gracey [phonetic] and this is the very first section
of Chapter 1 using the blitzer [phonetic] combo book for introductory and intermediate
algebra. So today, as you see, when you're done with your homework, you're going to be
able to evaluate algebraic expressions, translate English phrases into algebraic expressions,
determine whether a number is a solution of an equation, translate English sentences into
the algebraic equations and evaluate formulas. So let's warm up, first of all, by performing
these operations and simplifying. Now just to let you know these are a little, you know,
above your pay grade. We're going to be learning some of this stuff as we go along but we might
as well get started and challenge ourselves right off. So why don't you go ahead and pause
this recording and try these warmups, see how you do and then play it again to see the
worked out solution. Good luck. [Silence] All right so here we go. Remember that negative
5 cubed is negative 5 multiplied to itself 3 times so what we'll end up with here is
the opposite of negative 125 which gives us a positive 125. We'll be reducing that by
3 and then 8 times 9 is 72. It's got a negative attached so we'll have negative 72. And then
we will have 122 when we do that numerator over negative 72. And then slowing down this
is able to be reduced. So if you see here, we can certainly divide the numerator and
the denominator by 2 and then we will end up with 61 over negative 36 which of course,
you can bring that negative sign to the front and we would get negative 61, 36. And this
is the fully simplified expression. As I said this might look -- we're going to be learning
about these topics as we go through the chapter so if this seems a little mystifying to you
don't worry about it. The second one, remember that in the order of operations, division
is higher on the hierarchy than subtraction. So really we have to -- we could write it
like this, 16 fifths decreased by 1 but now what we need to do is we need to get a common
denominator so that we are combining like size pieces here. So we'll have 16 fifths
minus 5 fifths which will yield the result of 11 fifths. And we'll just go ahead and
leave these as improper fractions. All right so now when we go down to evaluating algebraic
expressions, we're getting started on the current material. So we can replace a variable
that appears in an algebraic expression -- By a number. The process is called devaluating the expression. So here is a first look at
our order of operations. You perform all operations within grouping symbols. So some grouping
symbols that you're used to seeing are parentheses, square brackets, stuff like that. And that's
usually what is done first so such as parentheses. Do all multiplications. Multiplication in the order in which they occur from left
to right. Whoops, sorry about that. And then the third step, do all addition and subtractions
in the order in which they occur from left to right. So here we go with this first example.
We want to find the mistake. You know a lot of times it's useful if evil math teachers
try to sucker you into problems and to doing things that look like they'll work out great
but don't work out at all. For example, if you look at this problem following order of
operations, really what should be happening first? You know, should this division be happening?
Should the multiplication be happening or should this addition be going on? So here
what do you think the mistake is? Take a look and then see if it matches up. All right exactly.
What you probably noticed here the person put 5 divided by a 5. They did addition before
a division or multiplication which was the error. So if we were writing out what happened,
this addition shouldn't have occurred. We should've divided first because really division and multiplication
are on the same hierarchy because division is defined by multiplying by the reciprocal.
So what should've happened first was this here the 2 divided by 5. It's all right. So
[inaudible] an example here. We're evaluating this following algebraic expression but they're
giving values. All right so here we go. Wherever we see an x in this expression, we're going
to plug in a negative 2 and see what value this expression takes on when x is negative
2. So 2 times negative 2 plus 25 divided negative 2 minus 1. So here x is negative 2. Wherever
there was an x, I plugged a negative 2. So this will yield a result of negative 4 from
the 2 times negative 2 increased by 25 all over negative 3 and this will be 21 divided
negative 3 which yields an end result of negative 7 and we're done. Okay, let's -- why don't
you try the next one? Notice there's two different variables. All right so go ahead and pause
this movie and try the next one. [Silence] All right let's see how you did. So in this
one wherever there's an x, we're going to plug in a 10. And wherever there's a y, we're
going to plug in negative 4. So we will have 6 times 10 minus 9 times negative 4 plus 1
all over negative 4 minus 10. So again, the y took on a value of negative 4. I've highlighted
it in the yellow. And x took on a value of positive 10 which I have highlighted in the
blue. So 6 times 10 yields a result of 60. Minus minus is plus so we're going to plus
36, plus 1 all over negative 14. And this will end up being -- bring that negative to
the outside and we'll get 97 fourteenths. And that is our end result. All right so next
step. Let's talk about some key words for addition, subtraction, multiplication, and
division. For addition, some words for addition are sum. Here's a hard one, plus. [Chuckles]
Increased by. More than. Now you have to be careful, it's easy to confuse more than, you
know, with an inequality which we'll talk about later. So subtraction, difference, less
than, decreased by, and simply minus. All right multiplication. Here this little tiny
word gets people all the time, of. Of means times. All right? Product, times, multiplied
by, twice, all of those are examples of multiplication. Division. It's always, you know, counting
how many of one quantity are in another. So quotient is a word for it. Divided by. Ratio,
these are all some key words for division. All right, next up, we're already on page
3. So here we go. Let's, you know, translate this English phrase into mathese [phonetic].
So write the English phrase as an algebraic expression. I'll start off, 6 more than a
number. All right so 6 more than a number we could write as x plus 6. Twelve less a
number. What do you guys think that would be? You got it. Twelve minus -- whoops. Making
up my own math a little bit. We got to represent it as a number. So 12 minus x, the number
can be x. How about 2 times the sum of a number and 5 increased by 9? So let's break this
up a little bit. We can start with the sum of a number and 5. We start the sum of a number
and 5. We can write that bad boy as x plus 5. Put that in parentheses. So this is that
portion of it. And then if we're saying 2 times the sum of a number and 5, we can simply
multiply that by 2. Remember if you have a number in one parenthesis that implies multiplication.
And then if we want to say increased by 9, you could go plus 9 for this last part, increase
by 9, plus 9 and all the pieces go together. All right so here we are with some equations.
It's very important to understand the vocabulary in math. I mean the vocabulary is the framework
of mathematics. So if you understand the language, you're going to understand how to work these
problems much better. So an equation is a statement that two algebraic expressions are
equal. So what symbol does an equation always contain? You got it, an equals sign. So this
is the equals sign. All right solutions of an equation are valued of the variable that
make the equation a true statement. To determine rather a number is a solution substitute that
number for the variable and evaluate each side of the equation. If the values on both
sides of the equation are the same, the number is a solution. So let's check out example
4. In example 4, we want to determine whether the given number is a solution of the equation.
So here we go. For this one, we have a variable x so we want to plug in 5 where we see an
x and see if it works out. So here 5 plus 17. Is that equal to 22? Well, here we get
22 equals 22. That looks true to me. All right. So here again we've got 5 in for x. We completed
the calculations and we got a true statement. So don't forget to answer the question, yes
5 is a solution of x plus 17 equals 22. So in your homework, they're just going to ask
you for a yes or no but I like to put the complete sentence here. All right in this
next one. X is now -- or z rather is the variable and it's taking on a value of 8. So 5 times
8 is that equal to 30? Well here's the value for our variable. We plugged in 8. So 5 times
8, don't use your calculators. Just use your brain. It's 40, isn't it? And 40 is certainly
not equal to 30. So we would say no, 8 is not a solution of 5 times z equals 30. All
right so example 5. They want us to write each equation now as an English sentence.
So let's check out what we have going on. So I am thinking that we can use some of our
words here. So here we have a 9, right? So 9 less -- so the less is coming from this.
So 9 less is that portion. And then here as you see we're going to have 3 and that operation
is multiplication. So 3 times a number. In mathese, the equals sign means is. All right?
So the math version is equals, English version is. So we have the 9 less, the 3 x can be
read as 3 times a number. Is is the green [phonetic] and then 7, 7. All right. Now why
don't you try number 2. See how you do. [Silence] All right so here we've got this quantity
and the parenthesis. So we'll start with our 2 times the sum of a number and 5 is a number
-- actually we should probably be specific because it's the same number. So we should
probably say is the number less 4. So let's match it all up. And I got to put 2 times
not 2 time. So the sum of a number and 5, is this part of it? And then here, we have the doubling. So
2 times and then of course, equals translates to is and this last part, [inaudible], we
got to get -- you know this color is a little dark. So we'll go with -- let's see how this
one here works. So this here the number less 4 is that portion. So just so you know if
you put all of the pieces together you get the end result. All right next example, whoops,
we have the direction. We want to write each sentence as an equation. So here we go, the
difference between 40 and a number is 10. So when it says the difference between and
then it says the two numbers, you put them in that order and you put that operation in
between them. So we put 40 and a number different is subtraction, is equals 10. So you want
to break it down a little. Whoops. The difference between 40 and a number is this part. And
then is 10 is that part. You try the next one. [Silence] All right let's see how you
did. The product of 6 and a number. So 6 times x how that one translates. Increased by 3.
So we're adding 3. Is 33. There we go. Okay. Formulas and mathematical models. One aim
of algebra is to provide a compact symbolic description of the world. These descriptions
involve the use of formulas. A formula is an equation that expresses a relationship
between two or more variables. The process of finding formulas to describe real world
phenomena is called mathematical modeling. Such formulas together with the meaning assigned
to the variables are called mathematical models. All right so our last example for this section.
A bowler's handicap which is denoted by age is often found using the following formula.
Age equals 8 tenths times the difference of 200 and a where a is the knowst [phonetic]
the bowler's average score. So just, you know, basically for a handicap you want to kind
of even things up. All right so if your average bowling score is 145, let's figure out your
handicap. All right. So here, h, remember is equal to zero point eight times 200 minus
a. Well a was the average score and what was the average score? It was 145. So remember
here, a was the average score. Our average score was 145. So this will be h is equal
to zero point eight. We need to do the parenthesis first so that'll be times 55. So h will be
44. So the handicap would be 44. So what does that mean? Well your final score, if you scored 120 in
a game, bowled 120 in a game, you would take that 120, you'd add to that the handicap.
All right? What you got for the handicap and you would end up with your adjusted score
of 164. So remember the 164 is what we got for the handicap. I apologize. The 44 is what
we got for the handicap and when you add that to your average score, you get your final
score. And that's it. You just finished your very first section. Have a fabulous day.