There are two things written on the board, one on
the left, one on the right.
What do we call the thing on the left?
An expression. And what do we call the thing on
the right? What
is
an equation? Anybody know what an equation or what's the
definition of an equation is?
Take a wild guess.
Yeah. Like two expressions brought together by an equal sign.
Two expressions brought together by an equal sign So
we definitely have one expression another expression then like an
equal sign in between okay, we're close Would this be
an equation definitely not Will this be an
equation? Definitely not. We definitely need an equal sign.
And we need an expression on the left and an
expression on the right.
Here's the actual definition.
An equation is a mathematical statement that
two expressions are equal.
So an equation is a mathematical statement
that can be written in many different forms.
So why is this an equation?
Whoever said it was.
Why is it an equation?
Because it's saying that 2x minus 3 is equal to
5. Yeah, it's a statement that this expression is equal
to this expression. Is this an equation?
Yes. Yes.
this an equation? see everybody's thumb thumbs up yes thumbs
down no
they're
definitely not equal we agree yeah is
this a
mathematical statement
that
two Mr. Wittman is 19 years old is
that a statement?
That true statement No definitely
not Okay, so is this a
statement that? One expression is equal to another expression.
Yes. Yes, this is an equation Okay
this is an equation and is the statement true
yeah yeah this is called a true equation
this
is called a false equation Is this equation red?
Is it true, false, or neither?
It depends on what the value of x is.
If x is 3, it would be a true equation.
If x is any other number, it would be a
false equation. So this is neither true nor false.
this is called an open equation.
Basically, anytime you see a variable inside an equation it's
an open equation. All equations are
either true, false,
or open.
This is very important.
Is this an equation?
Yes, yes X is an expression X is an expression
is a statement the two expressions are equal
This contains a variable however.
This is the rare case that this is not an
open equation. This would be a what?
So but most any other time
like this equation is no longer true always it's
open okay but almost every time except for special
cases if an equation contains a
variable it's an open equation Can
Can we solve these?
7th grade? 8th grade?
Something like that.
What would be the first step?
Subtract 5 both sides or add negative 5 both sides.
Which property allows me to do that?
Either the addition property or the subtraction property.
If you add a negative I just call everything the
addition property of equality.
And then what's the next step?
Divide by 2. Okay,
let
me come back to
this. So if the directions were
solved,
what does solve mean?
Make the statement true.
the statement true. We'll come back to that in a
second.
Some people said find the answer.
What else did we say?
Find the unknown value.
Okay, it's a possibility.
So let's write down the definition.
to solve
means
to find
the
solution
set We'll come back to what
solutions that means in a second.
What uh, what is the solution to this equation?
X equals 3. Uh, it is not x equals 3.
It is 3. 3
is the solution.
So what's the mean to be a solution?
OK,
this is very important.
Here's the definition. a solution
is
a number
that
when replaced
for the variable
it makes
the equation
true
OK, so why is the number 3 a solution?
Because when we replace it with a variable,
we no longer have an open equation, we have a
true equation.
OK, so a solution is a number that when replaced
with a variable, it makes the equation true.
Why is the number 7 not a solution?
Because when it replaces the variable, it does not make
the equation true. Okay.
Okay.
Solution set,
the definition is
a solution set
is the set of
all solutions. So
we still haven't solved yet because we haven't found the
solution set. Well, we kind of did, but
our final answer has to be a solution
set.
Is x equals 3 the solution?
Why is x equals 3 not the solution?
It's still an open equation a solution is
a number not an equation, okay?
So the solution is
is three and the
solution set
you
simply take all the solutions there's only one of them
here and we put it in these curly brackets
there's several ways to
that denotes sets in mathematics.
If we could list them all, we simply use curly
brackets and put them all in there separated by comma.
There's only one here.
Are these three the same equation?
They're definitely not the same.
This is a statement that this expression is equal
to this expression, which this expression is way different than
this expression, and this expression is way different.
And this is way different than that one.
They're all different equations.
However they all have something in common.
What do they have in common?
The same exact solution set.
Okay. Like the addition property of equality says
you can add the same number to both sides of
the equation and the new equation is
equivalent.
Not the same equations, equivalent.
So these are all called equivalent equations.
Equivalent equations
are
equations. that have
the same
solution
set.
Another definition. Okay, so we use the addition property of
equality to go to here.
Here we use the division or multiplication property of equality
which says you can divide both
numbers by the both sides of the equation by the
same number and the new equation, it one's not zero
is equivalent which means that is the same solution set
okay and once we're here we know the obvious
solution is three can we plug in three equals three
okay
let's try another
one The directions say solve.
What does it mean to solve?
Find the solution set.
And what is the solution set?
The set of all solutions.
You take them all and you put them in curly
brackets.
Okay. And what is the solution?
A number that when replacing the variable makes the statement
true. Makes the equation true.
A number that when replaced for the variable makes the
equation true. A number that when replaced
for the variable makes the equation true.
So solutions, okay first step here is
Add eight to both sides Next step
Divide by five or multiply by one fifth same thing
Is this my solution
No, we're getting better, I like this.
What's my solution? Twelve fifths, or two point
four, something like that.
And what is the solution set?
Twelve fifths. So the way we say it, we say
the set containing
12
fifths.
We write down. It's just curly brackets and put the
number inside And why is 12 fifth the solution?
Because when replace for the variable or plug into the
variable it makes the equation true Why is the number
zero not a solution?
Because when you replace a
variable it does not make the equation true.
Okay.
Let's do another one.
Okay, here we got a very well on both sides.
So what's our first step?
extract 3x both
sides And
we have four equals seven.
What does that tell us?
It's a false equation.
Good.
So we use the addition property of equality.
We end up with a false equation, always false equation.
So what is the solutions?
There's none.
So is there a solution set?
No.
I'll leave it. there is.
Okay, it's simply
this set.
The set contains all of its solutions.
There are no solutions to this equation, but there is
a solution set. So can we even solve this
equation? What does it mean to solve?
Did we find the solution set?
Yes, we can solve this equation.
We can't find any solutions but we can solve it.
Ok, because by the definition we found the solution set.
In fact this is a very important set.
Its own name is called the empty set And in
fact, it's so important we give it its own symbol
Okay
It's a circle with a slightly diagonal line through it
This is also the empty set
You could either write this as your solution set or
you could write this as your solution set.
Simply an empty set.
In fact, it's one of the most important sets in
all of mathematics. Is this the empty set?
Definitely not, this is not empty.
It contains another set.
The set containing the empty set is definitely not the
empty set. So don't ever write this.
Terrible.
You can either write this or write this.
Don't put one inside the other.
Okay?
Okay. Let's try a new equation.
This has lots of variables, what do we get first?
I like my terms like that
And now what You could either stop here
This is always equal to this If you want to
we could subtract 4x we get this equation
and it doesn't matter if we do that or not
and what do we end up with?
an always true equation so what are the solutions?
Every single number we know is a solution.
And what is the biggest set of numbers we know?
All real numbers. Why is every real number a
solution? Because when you plug it in for the variable
it makes the variables in this case makes the equation
true. Okay, what is the solution
set? Okay, the solution set the set of real
numbers. Okay.
And there's lots of different ways we denote the set
of real numbers
One way is we write it out in words like
that
Another way is It looks like this have
we seen that before
Okay
That is simply the letter R.
But in boldface
That's not IR it's bold face
are Right when you're in Microsoft Word or whatever how
do you make something bold?
We highlight it and press bold
Okay Before computers you couldn't do that.
You had to make it bold by hand and so
back who knows how long
Okay, they made it bold by making the left side
Put a line there, okay, that's simply bold face
are It's just that is the one way we denote
the set of real numbers.
We're gonna learn another way.
Some of you might know this
we do this all the numbers between negative infinity and
positive infinity okay we'll typically do this way this is
called interval notation how many of you guys learned interval
notation last year
and not too many we'll learn it uh beginning next
week i think
Okay, so some equations have one solution,
some equations have no solutions,
and very similar equation all of a sudden we have
an infinite number of solutions.
Alright, either way, any equation we can
solve even if there's no solutions.
Is this an equation?
Why is it an equation?
It's a statement that two expressions are equal.
As long as you see an equal sign and some
expression on the left and some expression on the right,
it's an equation. Anybody know a solution to
this? Just like say
like x can go y and y can go like
5.
Close.
So this is fundamentally different because there's two variables.
So we actually have to alter our definition.
Where's our definition of equation?
Here it is. An equation is a mathematical statement that...
Sorry.
Definition of a solution.
A solution is a number that when replaced for the
variable,
this is really the only definition for an equation in
one variable. Okay, when we have an equation in two
variables, it's a different
definition. Let's write down
definition.
A solution,
equation,
and two variables.
It's not a number.
We need two different numbers
okay and
right for example one comma five
somebody said or one for the X five for the
Y is the solution so we had to know which
number goes to which variable and the way we denote
that is an ordered pair.
So a solution to the equation in two variables is
an ordered pair
such
that
when
the numbers
are
replaced for
their respective
variables
it
makes the equation true.
So it's not a number, it's two numbers.
We should call it ordered pair.
Such that when the numbers are replaced for the respective
variables, the x for the x my life or why
it makes the equation true.
So 1 comma 5 is a solution.
Somebody give me another solution
zero
comma three well okay
interesting I like that one half four
three comma
nine
100
comma
203
negative 10 comma
negative 17
and
we could keep going how many solutions are there to
this equation? Infinite.
Okay there's no way we're gonna
write down all the solutions so we're not going to
do a curly bracket so we can't write
them all plan.
So how are we going to write down the
solution set? Any guesses?
Okay,
so I have a question.
So what would I write?
So all real numbers?
And the range is?
All real numbers.
So does that mean every ordered pair is a solution?
No. It's definitely not.
0, 10 is not a solution.
So we can't just write down the domain range.
We'll talk about what domain range means in a minute.
How am I going to do this?
Because we know lots of numbers that are not solutions.
Sorry, lots of ordered pairs.
This definitely is
a linear equation.
Okay, in fact, you've written down the solution set to
this many, many, many times.
How do we do it in eighth grade?
We
graphed
the solution set.
Okay.
We draw a coordinate plane.
So when you graph linear equations in two variables in
eighth grade, and you probably did a bunch last year
too, you're really just graphing the solution set.
All the ordered pairs that were replaced for the variable
make the equation true.
There's an This line is simply an infinite
number of ordered pairs Every single ordered pair on that
line is a solution all of them together is the
solution set
Okay We could have graphed the solutions
for these these are actually also called linear equations but
only in one variable.
We could graph it.
There's only one variable we need only one number line.
We could put a dot right here on 12
fifths and we graph the solution set.
We put a dot on every solution.
We don't do that because it's pointless.
I mean it's a lot easier to do this.
Okay we could have graphed this solution
set here and we
put a dot on every single solution which is all
written numbers but we're gonna do that okay
but we do graph the solution set to linear equations
in two variables it's it's really the only way to
write them all down there are some other notations that
we don't to worry about that right now
okay this tells you what is a solution what is
not a solution
okay
so
did we solve this equation
yeah why
do we solve it found the solution set.
Okay, this is very important.
We're always either going to start with an expression this
class or an equation.
Okay,
this very first equation,
sorry, very first Expression I wrote here.
Can we solve this?
Why can we not solve It's
it's not an equation to solve needs to find the
solution set a
solution set that all solutions a solutions a number that
replace for the variable makes the equation true There's no
equation
Expressions are neither true or false
Okay, just is what it is Okay, you cannot solve
this you can't solve this an equation okay,
so when we When you look at something around the
board it's either going to be an expression or an
equation
We do totally different stuff Almost always we solve these
Expressions we can do we can manipulate them You
could rewrite them.
You could factor out a number.
You can change it, manipulate it, simplify it, complexify it,
but you cannot solve it.
Very important. A new definition.
The definition of a relation is simply a set of
ordered pairs.
It's called a relation.
Is this a relation?
Yes. Yes. Yes, why?
Because it's a set of ordered pairs.
Curly brackets mean it's a set.
and the inside separated by commas are?
Ordered pairs. Ordered pairs, this set of order
of pairs.
Okay. This has a very clear pattern.
Well, it should be clear.
1, 3, 2, 5, 3, 7, what's the next one?
4, 9. 4, 9, so on and so forth.
If there's a clear pattern, you just put dot dot
dot, the ellipses.
Okay, so here this is a relation.
Why is this a relation?
Because it's a set of ordered pairs.
That's right Because there's the brackets and there's everything inside
is ordered. This has an infinite of ordered pairs in
it Okay Okay, there are three ways
to
denote relations.
First of
all, you list
the ordered pairs. Just like we did
here and here. Okay.
It could be in a table too.
Doesn't necessarily have to look exactly...
Okay.
You can list them like that in the correct...
This is called roster notation you put them
in a table you can go 1 5 negative 4
and put arrows
2
ok doesn't really matter as long as you listing your
ordered pairs. The second way what What else can we
do with ordered pairs?
Just like we did up here, we graphed them.
We graphed the relation.
Thank ya. Okay, it could be like this or like
we did earlier. We can have an infinite number of
ordered pairs.
That's typically when we graph a relation.
When there's an infinite number of them.
And then what's the third way?
Do you want to take a guess what the third
way is? This
Was the solution set this guy, right
Okay, so the third way is we write down an
equation
Okay However, an equation is not
a set of ordered pairs.
Okay, so if we write down y
equals 2x plus 3, this is not a set of
ordered pairs. But,
what about this is a set of ordered pairs?
The solution set. So in equation,
it's the solution set.
OK. It's the solution set that is the set of
ordered pairs. It is the relation.
So we'll call this a relation
Technically, it's not the solution set that this is the
equation
If the solution set is the set of ordered pairs,
okay, so those are three ways we denote relations
Is this an equation?
Wait, will?
Sometimes it's a yes.
Definitely not. Is that an equal sign?
No. No, definitely not an equation.
What do we call this thing?
An inequality.
Okay. Can we solve inequalities?
What's the need to solve?
Find the solution set.
Okay, it needs to find the solution set.
What's the definition of a solution to an inequality?
Actually let's go back.
what is an inequality
what
is
it
So, what is the definition of an inequality?
Umm, close. Not, well, not really.
A question?
Sort of.
It's almost the same definition as equation.
Or is equation? A mathematical statement that you use questions
or a less than variable?
Okay, so instead of equations, it's going to be an
inequality. We'll write this down in a second.
It's a mathematical statement that one expression
is either less than, greater than, less than or equal
to, greater than or equal to another expression.
Let me write that down.
community and inequality
a mathematical statement that one expression
is less than
or greater than or less than or
equal to or greater than or equal to another
I
could have wrote the words less than or greater than
or
less than or this is it gonna be a number
or an ordered pair
a number
what because only one variable
okay
how many
solutions does this have
okay
is zero a solution
yes is
Is 3 a solution?
Yeah.
Is 4 a solution?
Yes. Is 4.5 a solution?
Yes. Is 5 a solution?
Yes. Yes. Why is 5 a solution?
Because it has a variable 2.
When replaced for the variable, it makes the inequality true.
Because of the or equal to, this is true.
Okay. So it definitely has an infinite number of solutions.
We can write them in something called interval notation.
We'll learn that probably Monday.
Right now we can just graph the solution set.
Since there's only one variable, we need only one number
line. Okay, and you simply put a dot
on every single solution.
You did this last year, correct?
Solved inequalities. You typically graph a
solution set. Let's put negative nine.
What was our first step?
and then divide by 2 do I have to flip
the inequality sign?
No. No. Because
we're
not
No. No.
All three of these are called equivalent inequalities.
Okay.
So we're going to graph the solution set.
We've got to put a dot on every single solution.
Is the number 3 a solution?
Sorry, is the number negative 3 a solution?
No. No.
Okay.
We have to just remove one tiny point.
Okay, it's very hard.
If you pull that one tiny point, you'll never see
that it's gone. so we exaggerated it make a big
open circle even though
okay
again there's gonna be a better way to
do the solution set but for right now we're just
gonna graph the solution set okay
all these definitions are very important what's it
Equation means we could make the solve with a solution
set with a solution Put our equivalent equation
or equation true false and open
We got to memorize these Know them by heart real
quick
We will have a quiz probably in the next week
on the definitions