In this section, we cover a lot of fundamentals that are essential to your success in
algebra. So we're gonna go over these concepts again to give you a bit more
practice.
So the first concept we'll talk about is translating
English sentences into mathematical expressions.
Now, when you work with math, you deal a lot with symbols that represent
quantities, and sometimes those symbols can seem like a foreign language.
So we're gonna practice a bit more, and that will help with your fluency.
So the first sentence we'll translate is:
"the difference of 6 and x is y."
Now, these words in this sentence
all represent mathematical symbols, so we can look at
the first one: "the difference." Difference represents subtraction, so we know that
we're going to have a minus sign
in our expression, and we're finding the difference
of this quantity, 6 and x.
So we write that as: 6 - x.
Now, in this sentence, the word "is"
represents our equals sign, so we'll write that next
and then bring down our final variable, y.
So, "the difference of 6 and x is y" translates to
6 - x = y.
Let's move on to our second sentence--it's a little more complicated,
and has a few more components to it.
So we have, "half the difference of t and s is less than twice the sum of t and s."
Looking at this first word, "half," we're going to translate that into a fraction
and write 1/2. Now, this fraction
here, we're taking half of this quantity,
the difference of t and s, so that means that we're going to put some parentheses into
our expression. And inside our parentheses,
we're gonna write, the difference of t and s.
Now, "difference" represents subtraction, so we're going to
subtract t and s, so t - s
is our expression inside these parentheses. Now, moving on,
this phrase, "is less than" translates to
our less than symbol, <, and then "twice" we can represent
by the number 2, because we're going to double
the sum of t and s. Now again,
this is a quantity, so we're going to use some parentheses again, and
inside our parentheses, "sum" represent addition,
so we're going to add a plus sign, and we're adding t and s.
So t + s goes inside our parentheses.
So this sentence: "half the difference of t and s is less than twice the sum of t and s"
translates to 1/2(t - s) < 2(t + s).
So let's move on, and
we'll work with simplifying and using our order of operations.
Now, the order of operations is a very important strategy to use when working through math problems.
I'm going to walk you through a few problems, and you'll see that although they have some
similar numbers,
the order and the grouping of their operation symbols is different,
therefore they're going to have different answers. So let's review the steps
to the order of operations. Our first step is to work through
all the math inside a set of parentheses. The second step is to work with exponents,
the third is to work the multiplication and the division present in the problem,
and the fourth step is to work the addition and the subtraction.
Now, another important thing to remember when using the order of operations
is that you move from the left side of the problem to the right side
of problem as you do your math within any of those steps.
So first, let's try this problem here:
we have 2 + 3 • 2² + 3².
Now, working with our order of operations, we're looking for parentheses
first; now in this problem, there are none,
so we're going to move on to our second step, which is working with our exponents.
So we have: 2 + 3 • 4 + 9.
Now we can move on to our third step in the order of operations, which is working the multiplication
and the division so we're gonna bring down our 2 we're not going to work
with that just yet
we're gonna find the product of 3 and 4 which is
12 and then bring down our 9. Now we can move on to
our final step of the order of operations and work our addition
again we're gonna move from the left side of the problem to the right side of the problem
so we're gonna 2 + 12 = 14 and then 14
plus 9, which is 23. So, 23 is our answer
when we simplify our first problem. The second problem here we have 2 + 3 times the quantity
2² + 3²
we're gonna look inside our parentheses first and work this math
so we have 2 + 3 we're not gonna work with that just yet
inside our parentheses, then we can use our order of operations
within these parentheses and we're going to work our exponents before we find the sum.
So we find 2², which is 4.
Plus 3², which is 9. Now, we're still working inside our parentheses to simplify this
as much as
we can so we have 2+3 times the sum
4+9, so we're gonna add these, which is 13.
Now we can move on we've simplified everything
within our parentheses, so we'll move on down the list and work our
multiplication, so we have 2 + 3
2 + 3 (13)
is 39 and now we can do our final step of addition
2 + 39 = 41
there's our answer for our second problem. Let's do our last one
here we have (2 + 3) and that sum
we are multiplying by 2² + 3²
so we're going to work inside our parentheses first moving left to right
we're gonna work our first set parentheses first
2 + 3 = 5.
Now, in our second set of parentheses we are gonna move down the list and do
our
exponents first again, before we do our sum
4 + 9
and now we can simplify within
this set of parentheses here, so we have
5(4 + 9)
is 13 and now we can do our multiplication, so
5 (13) = 65
Now we'll move on and talk about some properties that
will help you simplify some expressions.
When simplifying expressions it's important to become comfortable with
three main properties. The first of which is the commutative
property and this works for addition and multiplication.
So, for addition this property states that you can change the order any two numbers in a
Sum and not change the result, so we have
A + B is the same as B + A.
Now, for multiplication the same thing goes. You can change the order of
two numbers in a product and not change the result as well.
So, we have A • B is equal to
B • A. Now, our second property here is called
the associative property and this says that you can change the grouping
within a sum and not change the result, so we have
A + (B + C)
= (A + B)
+ C and then the associative property for multiplication says that you can change the
grouping within a product
and not change the results as well. So we have A
A (BC)
A (BC) = (AB) C.
Now, our third property is called the distributive property
and this states that multiplication distributes
over addition. So if we have a number outside a sum here
our number is A, this will distribute over
each term within this sum. So this A (B + C)
becomes AB + AC.
So let's work through a couple problems here and see how these properties apply.
For our first problem we have 3 + 4
3 + 4 (5A + 3)
3 + 4 (5A + 3) + 4A, so we're going to use our order operations
as well as these properties to simplify this expression.
So first off let's look inside our parentheses. Now these two terms are not
similar, so we can't simplify that
any further and we don't have any exponents to work, so let's move on to multiplication.
Now this is where our distributive property comes in. So we are going to
distribute
our 4 over each term inside these parentheses.
So let's bring down our 3 cause we're not gonna work with that just yet. We're gonna
distribute our 4 over the 5A and the 3
so 4 (5A) = 20A
and then 4 (3) = 12, so we're gonna add 12
and then bring down our final term, which is 4A.
So now we're gonna use our commutative property for
addition to rearrange some of these terms
and make them easier to work with. So as we can see
we have to whole numbers and two terms that include a variable
so let's move our terms that have a variable
over here. We have 20A and then lets bring
4A over here as well
and then we'll bring our 3 and our 12 over.
So now since we've rearranged these terms we can apply the associative property
and group them
in order to simplify. So let's group our terms that have our similar
variable and group our whole numbers and then we'll add.
So we have 20A + 4A
20A + 4A = 24A
and then 3 + 12 = 15.
So our final answer
simplifying this expression gives 24A
24A + 15. Let's try another one. So here,
we have 4 (2 - 6x)
4 (2 - 6x) - (3 - 4x)
so the first thing that i'm gonna do here, is i'm gonna change all my subtraction
signs
into additions plus a negative. So here we're gonna
add -6x. This subtraction sign here is
as if we were adding a -1 times this
quantity and then this subtraction is adding
-4x. That will make the math a little easier to work with.
So now let's first start we're gonna work from left to right here
and let's use our distributive property to distribute
this 4 over each term inside
these first parentheses. So we have
4 (2) = 8 and then we're gonna add
4 (-6x)
which is + -24
+ -24x. Now looking
over here to our second set of parentheses we have a -1
-1 here that's gonna distribute over each term
so we're gonna add -1 (3)
-1 (3) = -3 and then -1
-1 (-4x)
-1 (-4x) = 4x.
Now just as we did in our first problem let's rearrange some of these terms to
make the math easier to work with and combine then simplify
so we'll move our -24x
first and let's bring our 4x over right next to it
and then we'll group our whole numbers here
8 and our -3.
So now let's group these using our associative property.
So we're gonna add this sum and this sum
So -24x + 4x
gives us -20x and then we're gonna
add 8 + (-3)
which is 5. So this is our final answer for our second problem -20x
-20x + 5. So it's also important to note
that you could rearrange these terms according to our commutative property for
addition
and still have the correct answer. So we could write
5 - 20x
and these two expressions are equal.
So let's move on to our final concept and we're gonna talk about sets.
When working with a set you're working with a collection of things
and in our case those things are numbers or variables
those numbers or variables, they are called elements.
We can write the elements of a set within a set of
brackets. We can also write an element of a set
using this notation.
Now this symbol here represents that x is an element of set A.
So this is important notation that you should remember when you're working with sets.
The other symbol that we'll use if we deal with a set that doesn't have any
elements to it we call it an empty or a null set and we
use this symbol. So this symbol here represents an
empty set
which is also called a null set
and that means that it's a set that doesn't have any elements in it.
Now we'll use two basic operations when dealing with sets
the first of which is a union. The union
means that for set A and set B
we're finding all the elements that are contained in
either one or the other, so the word
"or" here is an important one to remember.
We can write the union of set A and set B
using this symbol here. So this means
that all of the elements in set A or
all of the elements in set B are what we're considering.
So this is the union of set A and set B.
Our second operation is the intersection of two sets
and the operative word here is "and". So when we talk about the intersection of two sets
we're looking at the elements that are contained in one set
and in the other set, so they're in both and we use
this symbol to denote
an intersection. So this is the intersection between
set A and set B. Now the final notation that we'll discuss here is called
set builder notation and this is a way to kind of summarize all the elements that
you're considering when you
deal with two sets. So let's use the union of set A
and B as an example. So we'll write that like this
and we can write the union of set A and B is
equal to all of the values for x
such that this vertical line here represents "such that"
x is an element of A
here's this notation here
or because we're dealing with a union we're gonna use the word
"or" x is an element of set B.
So there's your set builder notation.
Let's try an example. So let's consider set A
includes the elements 0, 2, 4, and 6.
Set B includes the elements 1, 2, 3, 4, and 5.
And set C includes 1, 3
5, and 7. What if we needed to find the elements
in the union of set A
and the intersection of B and C?
Now we can use our order of operations and work inside our parentheses
first. So let's find the elements that are contained
in the intersection of set B and C.
So, here, if we're dealing with the intersection
we're looking for all of the elements that are contained in both
sets. So looking up here for set B
and then comparing it to set C, we have 1 in both sets,
2 is not, but 3 is in both sets.
4 is not, 5 is in both sets
and 7 is not. So this
set is 1
3 and 5.
Now let's bring down and compare this
found set that we just did to the numbers that are the
elements in set A and we need to find the union of
set A and these element here.
So when we deal with union we're dealing with the numbers that are in
either set A or the second set.
So, if we look at set A, we have the numbers
0, 2, 4
and 6. So our
elements for this notation here
are 0, 2, 4, and 6 or 1, 3, and 5.
So we can combine these, and write
{0, 1, 2, 3, 4, 5}.
So, these are all of your elements
that are in the union of A, and the intersection of B and C.