Hey, guys.
Welcome to Algebra I.
Today we're going to be working
on evaluating expressions.
To help us get a grip on this,
we'll journey back
to the order of operations.
Ready? Let's go.

(Describer) She uses a stylus on a screen.

All right.
Order of operations.
Think I skipped one.
There we go.
When you learned
the order of operations,
I bet somebody somewhere
taught you this acronym:
PEMDAS.
Remember what it stood for?
I bet you do.
Let's switch to
that pointer tool.

(Describer) P-E-M-D-A-S.

P stood for parentheses,
then you got your exponents,
multiply, divide,
add, subtract.
That told you the order
that you had to work
to simplify an expression.
For example--

(Describer) Six minus 2, plus 3 squared.

Let's get the pen back.
So, I look at
that numeric expression,
and I see I've got
some parentheses.
The order of operations says
if I see parentheses,
whatever is in the parentheses,
I handle that operation first.
I've got 6 minus 2.
6 minus 2 is 4.
I'm gonna do some mental math,
but feel free
to use a calculator
to make sure
that you're right.
Or you can do
this mental math with me
and get your skills stronger.
So, 6 minus 2 is 4,

(Describer) She writes.

and I'm not gonna handle
that 3-squared yet.
Just keep writing that down
until you're ready
to deal with it.
Then order of operations says
I would do exponents next.
So, I see I have 3-squared.
3-squared is 9,
because you remember when you
raise a number to a power,
you're just multiplying it
times itself.
So, 3 times 3, that is 9.
So, that expression
would simplify to 4 plus 9,
and I've handled my exponents.
I don't have multiplication
in my problem.
I can check that off.
There's nothing to deal with.
I don't have division
in my problem,
so there's nothing to deal with.
But I do have some addition.
4 plus 9 is 13.
I've handled that, and I'm
down to one thing. I'm done.
That answer would be 13.
I didn't need to subtract.
But somewhere along
your math career,
you handled problems like that
using order of operations.
Somewhere along the line,
PEMDAS changed to GEMDAS,
because the math people
didn't want you to think
that parentheses
were your only types
of grouping symbols.
Because sometimes,
like in this problem,
you don't have parentheses,
but you have an absolute value.
Or you may have braces
or just anything
to group those items together.
On this one, I see
I have absolute value.
I have 9 plus
the absolute value of 3 minus 4.
Those absolute value bars
are a grouping symbol,
and I have to handle
the operation
that's going on in there first.

(Describer) The bars are vertical.

Let's switch to our pen.
So, the absolute value
of 3 minus 4.

(Describer) She writes.

Absolute value
is a way to describe
how far away
a number is from zero.
In this case, first handle
that operation inside,
and then we're gonna take the
absolute value of that number.
So, 3 minus 4.
That's -1.
And the absolute value
of -1 is 1.
-1 is just one unit away
from zero on the number line.
A shortcut way
to think about absolute value
is just whatever
your number is inside,
make it positive
if it's not already.
That's a shortcut way.
For this one, now that I know
that the absolute value
of 3 minus 4 is 1,
I'll insert that
back into my problem.
Scoot that over,
give myself more room to work.
So, the absolute value
of 3 minus 4 is 1.
So, 9 plus 1.
So, I've handled
the grouping symbol.
I don't have any exponents.
I don't have anything
I need to multiply.
Don't have anything
I need to divide,
but I do have some addition.
So, 9 + 1. That's 10.
You're done with this problem.
You're down to one number.
There's nothing left
to subtract.
You've simplified
that expression:
9 plus the absolute value
of 3 minus 4 is 10.
Let's keep going.

(Describer) X-squared plus one.

So, here we go.
Now, this expression
is not numeric.
It's algebraic.
How you know what the value
of this expression is
all depends on what number
you let take the place of x.
For example,
let's say I replace x
with the number 3.
If I let x be 3,
let's see what the value
of this expression is.
I'm gonna replace
that x with 3.
I'd have 3-squared plus 1.
I'd keep going
with my order of operations.
Don't have any grouping symbols,
but I do have
an exponent to handle.
I've got that 3-squared.
That's 9.
9 plus 1. Going down
my order of operations.
I don't have anything
to multiply or divide,
but I do have things
to add together.
9 plus 1 is 10.
If I let x
take the value of 3,
then that expression
has the value of 10.
What if I didn't want it
to equal 3
and wanted it to equal 4?
How would that change
the value of this expression?
Let's get this out of our way.

(Describer) She erases her writing.

Let's see what happens
if x has a value of 4.
If I let x be 4,

(Describer) She writes.

then I'd have
4-squared plus 1.
Right?
All right, so 4-squared.
That means 4 times 4.
So, that's 16.
So, I've got 16 plus 1.
That's 17.
So, if x equals 4,
then that expression equals 17.
I've got to set up
this pattern for you
so you get a handle on this.
Let's say x is 5.
How does that change
the value of this expression?
If I let x equal 5.
Now what does
this expression equal?
I'd have 5-squared plus 1.
5-squared.
So, that's 5 times 5.
So, that's 25 plus 1.
25 plus 1 is 26. Okay.
So, depending on
what you replace x with,
the value
of your expression changes.
That's how you're gonna be
working through this topic
on evaluating expressions.
The value of the expression
depends completely
on the value of x.
You've got to know
the order of operations
in order to work your way
through these problems.

(Describer) She changes the screen. Titles: Example 1: Evaluate 5x-squared plus 1, when x equals negative 3.

(female describer)
She changes the screen.
Example 1: Evaluate
5x-squared plus 1,
when x equals -3.
All right.
Let's move along.
I'm gonna give you
a hint with these.
Don't try to memorize
every step to what I'm doing.
There are
a gazillion math problems,
and you can't memorize
the steps to all of them.
You'll be very frustrated.
I bet that
you've had a moment
where you take
all the notes in class,
you look at your assignment,
and you're like,
"We didn't do problems
like this."
You're never gonna see
exactly the same problem.
It's the process
that you've got to master.
Don't try to memorize
how I did these problems.
Memorize my process.
Look at, "I'm gonna replace x
with some number
and then use the order
of operations to solve it."
Just keep that process
in your mind, okay?
All right, let's keep going.
"Evaluate 5x-squared plus 1
when x equals -3." Okay.
So, this problem's telling me
that I need to replace x
with -3.
Let's do that.

(Describer) She writes.

Five. I'm gonna put -3
in the place of x,
and I'll put that number
in parentheses.
Get in the habit
of doing that.
It helps you get through
the problem easier.
I see I've got plus 1.
Now I'm gonna go to the order
of operations, grouping symbol.
I see I have parentheses,
but there's no operation
in the parentheses.
It just has that number
in there.
I can keep going.
Exponents.
I do see an exponent.
I need to go ahead
and handle that first.
I'm gonna scoot this problem
to the left
to give me space.
Okay. Off to the side,
I'm gonna handle
that -3 to the second power.
If I raise a number
to the second power,
that means I'm multiplying it
by itself.
So, -3 to the second power,
all that means
is -3 times -3.
Okay? So, remember
your integer operations.
When you multiply
two negatives together,
your answer's always positive.
So, a -3 times a -3
is a positive 9.
Okay? So, I did that work
off to the side.
Now I'm gonna get back
to my problem.
Looks like I left my + 1
hanging over there. Okay.
Now that I know...
that -3 to the second power
is 9,
I'm just gonna go ahead
and replace that in my problem,
'cause I know
that has a value of 9.
Okay? I've taken care of
the exponent.
I'm gonna keep working
my way down.
Do I see any multiplication?
Yes, I do. I have 5 times 9.
I'll handle that.
That's 45 plus 1.
I've taken care of
the multiplication.
Do I see any division?
I don't.
Do I see some adding? I do.
So, 45 plus 1.
That's 46.
And you're all done.
You've made your way to the end.
So, when you evaluate
that expression
with x equals 3, you get 46.
Okay? Let's keep moving.
Remember, just replacing that x
with whatever value I'm given,
and then using
the order of operations.
All right, Example 2.

(Describer) Evaluate 3 minus 10x, when x equals two-fifths.

(describer)
Evaluate 3 minus 10x,
when x equals 2/5.
Now, I know
you saw that fraction,
and depending on
how comfortable you are,
you either cringed
or you're like, "I can do it."
I hope it was the second one,
but if not, it's fine.
"Evaluate 3 minus 10x,
when x equals 2/5."
All right.
What do you do first?
Remember, just replace x
with what you've got.
So, if x is 2/5,
I'm gonna replace x with 2/5.

(describer)
She writes 3 minus 10
times 2/5.

(Describer) She writes 3 minus 10 times two-fifths. Two-fifths is in parentheses.

2/5 is in parentheses.
Now, off to the side,
I'm gonna handle
that multiplication.
Let me check off "G." I've
handled the grouping symbol.
I don't have any exponents.
I'm at the multiplication step.
Was getting ahead of myself.
So, I'm gonna show you
a trick of how to handle
this multiplying
with this fraction.
You have 10 times 2/5.
I'm gonna use the dot here
instead of the parentheses.
I'm gonna switch to the dot.
Any whole number
you can make a fraction
by putting it over 1.
I'm gonna write 10 as 10/1.
So, I really have
10/1 times 2/5.
And once you've made
your way here,
you multiply straight across.
I'm gonna scoot that over
so you can see all of my work.
Okay.
I'm going to multiply
straight across here.
So, 10 times 2.
That's 20.
1 times 5. That's 5.
Now you just have
a simple division problem.
20 divided by 5.
Okay. That's 4.
What that means is,
this part of our problem
right here, that 10 times 2/5,
it just simplifies to 4.
So, I'm just gonna replace that:
3 minus 4.
So, that crazy fraction
just broke down
when you multiplied it by 10,
and you got 4.
You've handled
the multiplication.
You don't have any division.
You don't have any addition.
You do have some subtraction.
So, 3 minus 4.
That's -1.
And you're all done.
In that expression,
if you let x equal 2/5,
your answer's -1.
Okay? Let's do a few more.
"Evaluate x minus 6
divided by 8,
when x equals 13.
We have another fraction
in this case,
but this time our fraction's
the expression that we're gonna
plug something into.
Don't let it get you
all bothered.
We're gonna use our same steps.
I've got x minus 6
divided by 8,
and it's telling me
to let x equal 13.
So, I'm gonna replace
that x with 13.
So, I have 13 minus 6
divided by 8.
All right, let's keep going.
I don't have
any grouping symbols.
I don't have any exponents.
I don't have any multiplication.
I do have some division.
Now, let's kind of
back it up a bit.
You could think of this top,
my numerator in this fraction,
as kind of being a group.
It may help to think that way,
because you have to handle
that top part first.
I can't start dividing
until I simplify that numerator.
So, what is 13 minus 6?
It's 7.
So, I really have
7 divided by 8.
And in algebra,
we're not afraid of fractions.
So, you don't need to
divide that any further.
Your answer is 7/8.
Your answer
actually is a fraction.
Okay? Let's keep moving.
I hope you're getting
comfortable with fractions.
I'm gonna keep
throwing them at you.
"Evaluate the absolute value
of -7x plus 4,
when x equals -2."
Okay?
So, remember your process.
First, go ahead and replace x
with that value.
So, I have
the absolute value of -7
times -2 plus 4.
Okay.
In this case,
those absolute value bars
are my grouping symbol.
I need to simplify
that expression
within those
absolute value bars first.
Okay? So, now,
look within the bars.
There's a lot going on there.
You have to handle
the order of operations
even within the bars itself.
You have a grouping symbol,
the parentheses,
but there's no operation
in there.
It's just your number.
You don't have exponents,
but you do have something
you need to multiply together,
that -7 times -2.
So, off to the side.
-7 times -2.
Remember, a negative times
a negative, that's a positive.
So, this is positive 14.
I'm gonna bring that into
my problem. Let's rewrite this.
So, I have the absolute value
of 14 plus 4. Okay?
Let's keep going.
We handled the multiplication.
We don't have
any division in there.
We do have some addition.
So, 14 plus 4.
That's 18.
So, I have
the absolute value of 18.
We said absolute value,
as a shortcut
you can think about it as,
"Whatever number's in there,
make it positive
if it isn't already."
The absolute value
of 18 is 18.
That means your answer
to your problem is 18.
That problem
started out large at first.
When you let x equal -2,
it breaks down
to the answer being 18.
Okay? I think
we've got two more.
"Evaluate x plus 4
divided by y minus 3,
when x equals -5
and y equals 6."
You see I threw you
a curveball a little bit.
On this one,
we have two different variables
and two different
"replacement values."
The numbers you use
in place of your variables.
are called replacement values,
'cause you're replacing
your variable with that value.
Don't let it upset you.
Just replace them one at a time.
We're gonna put them
in there. Okay.
If x is -5, that means
that my numerator,
I'd add have -5 plus 4.
So, I've got that one in there.
Divided by...
It said to let y equal 6.
So, 6 minus 3.
Okay, so I've got that
in my problem.
On that last one, we said
you could think about this
as two different groups--
your numerator being a group,
and your denominator
being a group.
You need to handle that first
before you get into your
division to simplify all of it.
Okay. So, -5 plus 4.
I'm gonna go across
this time with my work.
So, -5 plus 4.
That's -1.
Then, in my denominator,
I have 6 minus 3.
So, that's 3.
You remember we're comfortable
with fractions now, right?
So, you can just leave that
as your answer: -1/3.
And you're all done.
Okay? Last one
we will do together.
"Evaluate -4 times the absolute
value of x-cubed plus 1,"
all right, "when x equals 2."
We're gonna use
our same process.
Just start by replacing
that x with 2.
So, -4 absolute value--
Gonna use parentheses for my 2.
2 to the third plus 1,
absolute value.
Sometimes people ask, "How do
I know if I use parentheses
when I replace the value,
or do I always have to?"
Get in the habit.
Whenever you replace x with
some value, use parentheses.
It won't hurt anything
if you do.
Sometimes it will hurt something
if you don't.
Just get in the habit
of always using parentheses
when you use
your replacement values.
So, order of operations.
The "G" first.
I do have a grouping symbol.
That absolute value
is trapping that group there.
I need to handle
what's going on in there first.
I've got 2 to the third power.
So, off to the side.
Raising a number
to the third power
means we're multiplying it
by itself three times.
So, 2 to the third
is 2 times 2 times 2.
Okay? Then you
just simplify this.
So, 2 times 2 is 4,
and then 4 times 2 is 8.
So, 2 to the third power.
That equals 8.
Now that I know that
this part of my expression
simplifies to 8,
I'm gonna write that in there.
So, I have -4
times the absolute value
of 8 plus 1.
Okay? We've still got operations
going on in that group,
so we want to get that
worked down.
I don't have any multiplication
going on in there.
Don't have any division,
but I do have some addition.
So, 8 plus 1.
That's 9.
So, I have -4 times
the absolute value of 9.
All right.
So, off to the side.
The absolute value of 9.
Remember? Just make it positive
if it's not already.
It's 9.
Now I know that
the absolute value of 9 is 9.
Keep in mind that
you're multiplying -4 times 9.
At this point,
I'm putting the dot in there
to show that I'm multiplying
those numbers together.
So, -4 times 9.
Remember when you
multiply numbers,
if your signs are different,
then your answer is negative.
So, -4 times a positive 9
is -36.
And you're all done.
You have simplified
that expression.
All right?
Go to the next one.
We're at the point
where it's your turn to try.
Pause the tape.
Take your time and work
through these problems.
Remember, none of the problems
are just like
any of the ones that we did,
because it is the process
that you're gonna want
to follow to solve these.
Remember, replace x with
whatever value they gave you,
and then use
your order of operations
to work your way through.
After you've worked these out,
press play again,
and we'll compare our answers.

(Describer) Titles: Evaluate for each given replacement value.
Number One: 7 minus 9x, when x equals two-thirds.
Number Two: 12 minus 3x, all divded by 5, when x equals 4.
Number Three: 2 times the absolute value of x-squared plus y-cubed, when x equals 4 and y equals 1.

(describer)
Evaluate for each given
replacement value.
Number 1: 7 minus 9x,
when x equals 2/3.
Number 2: 12-3x all divided
by 5, when x equals 4.
Number 3:
2 times the absolute value
of x-squared plus y-cubed,
when x equals 4
and y equals 1.

(instructor)
Let's see how you did.
So, you evaluated 7 minus 9x,
when x was 2/3.
Let's switch
to that pointer tool,
move the answer box
out of the way.
You should have got 1
for that one.
"12 minus 3x divided by 5,
when x equals 4."
On that one,
you should have got 0.
This last one, you have
2 times the absolute value of
x-squared plus y to the third,
when x equals 4 and y equals 1.
Your answer for that one
should have been 34. Okay.
All right.
I hope you got a handle
on evaluating expressions.
Remember, replace the value
that they gave you for x,
and then use
the order of operations
to work your way through.
Hope to see you soon. Bye.

(Describer) Accessibility provided by the US Department of Education.