Transcript College Algebra I

College Algebra I
Week 1
The University of Phoenix
Inst. John Ensworth
Get ready for different kinds of
information
• Math is a language…a foreign language
• So you need to just plain memorize some of
the terms and concepts
• I suggest keeping a stack of note cards next
to you and write down definition words so
you can flash test yourself as the weeks go
on! For this first night, use the supplied
blank sheets of paper to note what you will
put to cards this week.
It’s not easy but…
• It IS very systematic. You can do the same
thing step by step over and over and get the
right answer every time.
• Expect to stretch your brain!
• The learning will go well IF you erase your
learned fears of math. Kids can get it --- so
can YOU!!!
Practice!!!
• So my suggestion is you do EVERY
problem in the book (well Chapter 1,2,3,7
for this part, part I).
• I’ll supply the worked answers for MANY
non-assigned problems on my Website.
• If you just don’t have time for all the
problems…at least copy the worked
problems onto a fresh piece of paper!
Pass it through your brain!
Chapter 1
• Definition words will be in Italics :
• Sets of numbers
– A collection of (a set of) counting numbers
(numbers you count with on your fingers).
Counting numbers are also called Natural
Numbers
– {1, 2, 3, … } For a set we (tradition) put
enough numbers in {}’s to give you the idea of
what is to come. The “…” means “and so on”
Sets of numbers we work with continued
• If you stick 0 into the Natural Numbers (which you
can’t count with your fingers) you get a set called
Whole Numbers (whole…whooohle….zeroooo)
• {0, 1, 2, 3, … }
• Why make a set with just a zero added?
It is the set of all numbers that can show up in a digit.
Why call it by a new name (Whole numbers?)
• Probably to tell you you have all the basic building
blocks. You can’t write “10” without whole numbers.
More cool sets…
• The Integers
• You need these to allow your accounting to
go into the red.
• It’s the whole numbers with all negatives
added…
• {…,-3,-2,-1,0,1,2,3,…}
• Integers are every single digit possible in
both directions off of and including 0
Expanding our set vocabulary
• Now to Rational Numbers
• (Different from irrational and imaginary… more
later)
• If you divide any two integers (as long as the
bottom number is NOT 0) you get a rational number
• Some Examples
• 31/6, 27/3, 5/1380, 1483949/2 and if you plug
those into a calculator, you get decimal rational
numbers
• 5.16667, 9, .003623, 741971.5 and anything else!
More on Rational Numbers
•
•
•
•
The set looks like some rules:
{a/b | a and b are integers, with b ≠ 0}
We can read this as :
The set of all numbers created by dividing a by b
such that a and b are integers making sure b isn’t
ever zero
• A neat foreshadowing is happening. a/b suggests
you can stick any integer into those letters you
like. They have a variable value, depending on
what you want to do. So they are called variables.
The Number Line
(seeing the numbers)
• Like a thermometer, we can string out the
numbers so we can SEE what is going on.
• This leads to make the number line.
-5 -4 -3 -2 -1 0 1 2 3 4 5
|----|----|----|----|----|----|----|----|----|----|
1 unit
the Origin
• The points on the line are called coordinates
(like addresses)
More on the number line
• What is bigger and what is smaller?
-5 -4 -3 -2 -1 0 1 2 3 4 5
|----|----|----|----|----|----|----|----|----|----|
Smaller numbers
larger numbers
Example 1 (pg4) – use a pencil!
• When we get to a place in the textbook
where it has examples, we’ll discuss them
and I’ll make up some more in the front.
• Pencil the answer once you understand why
it’s right. If you don’t get it, then get help
from your group members, me as I walk
around, or on breaks, or between classes.
Sometime! Get help.
**EX 7-18**
Example 2 (pg4)
• These examples combine the definitions of
the sets of numbers (integers, whole
numbers, rational numbers, natural (or
counting) numbers) with our number line.
• **EX 19-28**
The real numbers on a number line
• There is a place on a number line for every real
number as well, but you have to estimate
-7/3
½
7/3
pi
-5 -4 -3 -2 -1 0 1 2 3 4 5
|----|----|----|----|----|----|----|----|----|----|
A special note on pi –
a famous irrational number
• pi = 3.141592653589…
pi and other special numbers are ALSO
found on the number line, but they are
called irrational. Why? You can’t make
them using two integers divided by one
another. They don’t repeat or end…
Irrational Numbers
• Irrational Numbers are like real numbers
except the digits to the right of the decimal
point never end.
• More on that later
(along with 2 = e =1.414214…)
• You’ll return to these in the second part of
the course in – MTH209
Page 5 ‘who owns who’ in the
sets we’ve covered…
• Who belongs as part of who in the number
definition game?
Example 3
(pg 5)
• This works more with the definitions of our
number sets. If you can think of just ONE
exception, then the T/F’s are false.
• **EX 29-40**
How Real Numbers Look on the
Number Line
(pg 6)
• We want to mark out, on a number line, a
range of numbers. Simple – no?
• And we want to tell people if we are
including the end numbers or not.
• We use ( )’s to say everything between (but
not the end numbers).
• WE use []’s to say everything between
including the end numbers.
Example 4 (page 7)
• So the problem tells us the range of
numbers…and if we are using one or the
other end numbers.
• Then we color in a line with ()’s or []’s (or
one of both) at the ends.
**EX 41-46**
Example 5 (page 8)
• Of course we may not have a high or low
end number. It might just go on forever.
• We use the infinity symbol in the problem,
and draw the line off to that end and stick an
arrow on it to say: “and so on”  ∞
**EX 47-52**
Playing with Integers, Rational
and Real numbers… all of ‘em
• Remember, Integers are all the positive
AND negative versions of whole numbers.
• The absolute value of a number is it’s
distance from 0
• Simply, you drop the negative sign IF it’s
there.
• We use two vertical bars to show absolute
values
Absolute Values
• So if you have 6 and take the absolute
value: |6| = 6
Big deal.
• If you have –6 and take the absolute value:
• |-6| = 6 Some deal, we dropped the ‘-’ sign.
Example 6 (pg 9)
• These examples are having you practice
dropping negative signs off numbers.
**EX 53-60**
Opposites attract… or do they?
• The definitions keep rolling over you like
an avalanche … keep writing them down!
• The next one is related to the absolute value
and the number line, it is knowing what the
opposite version of a number is.
• The opposite is the one the same distance
from zero on the other side of the number
line
Opposites
• Just multiply the number by ‘-’ to get the
opposite.
• The opposite of 8 is –8, the opposite of
–10 is 10.
• There is NO opposite of 0. It is special and
doesn’t have a sign (good for party trivia).
• The opposite of an opposite -(-a) = a
– Two negatives kill each other
Example 7 (pg 10)
• This example has you practice with these
negatives and absolute values.
• Notice they are slipping the variable ‘a’ in
on you. Get used to thinking of a single
letter as a place holder for any number you
want stick in later.
**EX 61-66**
End of 1.1 Exercises
• Remember, I’m supplying answers to many of the
non-assigned problems on the Website in PDF
format. These are to be used ONLY in this class and
no further.
• You can print them, or use them on your laptop.
• Also, form your Learning Team Now!
• In this work break, go to the section that is most
confusing to you and work on it (getting group help
or my help).
Then move on to the next trickiest stuff.
Fractions Section 1.2
• We almost always, in life, want to say how
much we have or are missing out of the
whole thing.
• Setting up a ratio in our minds is a very
natural thing to do.
• “The trunk is half empty.” “We’ve use up
about 2/3rds of the butter.” “We were
missing about 1 in 10 students today in
class.”
Fractions
• From the text (page 14), if you eat 4 of the 6
pieces of pizza you’ve eaten 4/6ths
• BUT this is 2/3rds of the pizza. This is
easier to visualize than 4/6ths. So we like
to simplify fractions to communicate better.
• So we need to play with fractions to know
how to do this properly…
Things that equal 1
• A simple but very important idea is what
happens when you play with the number 1
• 2/2 = 1 5/5 = 1 1000/1000 = 1 c/c = 1
• If you multiply ANYTHING by 1 it is the same
as it was before
• 8·1 = 8 ½ · 1 = ½ etc.
NOTE: I’ll use “·” for multiply … not “*” or “x”
• Easy?
So you can keep multiplying by 1
all day long!
2/3 · 2/2 = 4/6 (our pizza example)
2/3 · 3/3 = 6/9
2/3 · 4/4 = 8/12
2/3 · 5/5 = 10/15
And on and on
*see top page 9 for this example not expanded out
Building up Fractions
• What we just did (multiplying by things
equal to 1) is called building up fractions
a/b = a/b · c/c (note b and c can’t equal 0 - why?)
So on to Example 1 pg 14
Example 1 pg 14
• To solve these, you just want to know what
form of 1 was secretly used to get the new
bottom number (the denominator).
• Was it 3/3? 5/5? 10/10? 44/44?
• Trial and error will work.
• Dividing the bigger denominator by the
smaller denominator gets the answer right
away.
**EX 7-18**
Going backwards… reducing
• OK, so we have large fractions and we want
to reduce it down to be as understandable as
possible, we want to PULL OUT things that
equal 1.
• Trick, try 2/2, 3/3, 5/5, 7/7, 11/11, 13/13 etc.
first. (Prime Numbers)
• So let’s REDUCE
Reducing
• 10/15
• We want simple numbers, no fractions after
we take out things that equal 1
• Can we take out a 2/2? No 5/7.5 yuck
• Can we take out a 3/3? No 3.33/5 yuck
• Can we take out a 5/5? Yes! 2/3 cool!
• 10/15 = 2/3 · 5/5 = 2/3 · 5/5 = 2/3
Example 2 (pg 15)
• 15/24
• Try 2/2  7.5/12 yuck
• Try 3/3  5/8 nice!
More on Example 2
• 42/30
• Try 2/2  12/15 nice! But we can go further
Starting again but with: 12/15
Try 2/2  6/7.5 yuck
Try 3/3  4/5 nice! That’s as far as we can go!
We took out a 2/2 and a 3/3. This is the same as
taking out a 6/6 which is what the textbook shows.
**EX 19-23**
Which takes us right to
multiplying fractions
• We can multiply fractions (or you can think
of them as ratios or even odds in gambling)
• If you eat a half of a piece of a 3 slice pizza,
how much did you eat?
• You ate ½ of 1/3rd of the pizza.
• 1/2 · 1/3 = (1 · 1)/(2 · 3)= 1/6
Multiplying fractions
• Multiply the tops and write the top
• Multiply the bottoms and write the bottom
• a/b · c/d = (a · c)/(b · d)
• Where b and d cannot = Zero
(why again?)
Example 3 (page 16)
• Find the product (multiply) 2/3 to 5/8
• 2/3 · 5/8 = (2 · 5)/(3 · 8) = 10/24
• THEN we can reduce it to something people
have a better feel for
• Try taking a 2/2 out…
 5/12 nice!
**EX 35-46**
A short cut!
• You can take a shortcut and kill numbers that
automatically equal 1 already. (This is reducing
before multiplying).
• 1/3 · 3/4 Can you see the 3 on top and the 3 on
the bottom? Isn’t that just 3/3? Can you take
them out and say you are multiplying by 1?
• That leaves 1/1 · 1/4 = ¼
Quick n’ easy!
Another short cut…
4/5 · 15/22
Glancing at it, you might notice that there are
5’s and 2’s that might kill each other on the
tops and bottom. But to see it, let’s expand
everything and cancel what we can…
More on that shortcut…
4/5 · 15/22 = (2 · 2)/5 · (3 · 5)/(2 ·11)
(2 ·2)/5·(3 ·5)/(2 ·11) The 2’s cancel
(2 ·2)/5 · (3 ·5)/(2 ·11) As do the 5’s
So we’re left with 2/1·3/11 =
(2·3)/(1·11)=6/11
Using this for units!
• What do you do when someone gives you
meters and you only understand feet?
• Or miles and you need kilometers?
• Or miles/hour and you need feet/second?
• Using the c/c=1 trick does the work for you!
What equals 1?
• 5/5 = 1 ok
• But what about 1 hour/60 minutes? That
equals 1!
• 12 inches/1 foot = 1
• 5280 feet/1mile =1
• And so on!
Oh the power!
Example 4 (page 17)
• The first wants to change 6 yards to ? feet.
• 6 yd = 6yd ·3ft/1yd = 18 ft
this equals 1!
**EX 47-58**
Fractions are numbers too
• Since they are numbers, we can divide them
(we just finished multiplying them)
• If you divide by a number, it’s the same as
multiplying by 1 over that number or
1/that number.
• 10 divided by 2 = 10/2 = 5
• 1/3 divided by 2 = 1/3 ·1/2 = 1/6
• It’s what you’ve already done!
Division of Fractions
• a/b divided by c/d = a/b ·d/c
• Flip the second fraction over and multiply!
Example 5 pg 18 (trying this)
• 1/3 divided by 7/6
• Is the same as 1/3 ·6/7=
(1 ·6)/(3 ·7)=(6/21)
[you can now take out a 3/3 and get 2/7]
• Or you could reduce it before multiplying
• Is the same as 1/3 ·6/7=
(1 ·6)/(3 ·7)=(1·2)/(1·7)= 2/7
• It doesn't matter WHEN you reduce, but it makes
the answer easier for everyone.
Example 5 part 2
• Now 2/3 divided by 5
• Is the same as 2/3 · 1/5
•  (2 ·1)/(3 ·5) = 2/15
• (No reducing makes the problem simpler!)
**EX 59-68**
Adding and Subtracting Fractions
• (We should really order pizza!)
• This is a simple idea as long as the pieces of
pizza are all the same size.
• If I ate 2/6 of the pizza for dinner, and 1/6
of the pizza for breakfast, then I ate
2/6+1/6 = 3/6 = 1/2 of the pizza
(reduce)
EASY!
Subtraction (same size pieces)
• Or if we start with 6/6 pieces of pizza,
then I eat 4 pieces… how much is left?
• 6/6 – 4/6 = 2/6 = 1/3 of the pizza
(reduced)
Addition and subtraction of same
size pieces
• So a/b + c/b = (a+c)/b
• And a/b - c/b = (a-c)/b
• Life is good!
Example 6 pg 19
• 1/7 + 2/7 = (1+2)/7 = 3/7
• 7/10 - 3/10 = (7-3)/10 = 4/10 =
(2·2)/(2·5)= (2·2)/(2·5)= 2/5
reducing by expanding so we can see the 2/2
**EX 69-72**
WARNING!
• If the denominators (the number on the
bottom) of what you are adding or
subtracting are not the same, you have to
work with it before finishing the problem.
• You can’t just add 1/6 to 1/5. This is apples
and oranges (basically). It’s a pizza divided
in six pieces and another only divided into 5
(how do you do that?!).
Warning part 2
• That denominator problem didn’t matter when
we were multiplying or dividing. Those
problem are nicer, that’s why the textbook
taught it first.
• Now we need to make the apples into oranges.
We want the denominators to be the same, or
common.
• That’s where we get the term common
denominator.
The least common denominator?
(LCD – not a drug)
• If we make the bottom numbers the same, we have
a common denominator and we’re back to a
simple problem…
• But what if that answer is like 6/10? We need to
reduce (take out a 2/2 and get 3/5) to be finished.
• Why not get the smallest common denominator
first, then we don’t have to reduce later!?
• That is why we care about the least common
denominator (LCD).
Example 7a (pg 20)
• 1/2+1/3
• The easiest trick here is to multiply by things
that equal 1.
• What do we use? How about the OTHER
fraction’s denominator?
(3/3) ·1/2 + (2/2) ·1/3= (3·1)/(3·2)+(2·1)/(2·3)
= 3/6+2/6=5/6
We didn’t have to worry about the Least
Common Denominator (LCD) this time
Example 7b
• 1/3 – 1/12
• Hey, we can make the first number have a
12 in the bottom if we multiply that fraction
by 4/4. Is that ok? It sure makes life easier.
• (4/4) ·1/3- 1/12 = (4 ·1)/(4 ·3) –1/12 =
4/12-1/12 = 3/12
We can reduce after the fact by taking out a
3/3 and get = 1/4
Example 7c
• 7/12+5/18
• Now let’s try to find the LCD and make our life easier
later (no reducing in the end… we hope)
• Let’s find the multiples of the first denominator
3,4,6,12,24,36 etc…
• And the second 2,3,6,9,18,36 etc…
• We can use 36. Both numbers have that multiple in
common.
• So we will multiply both fractions by something equal
to 1 that makes the bottoms of both equal to 36
(3/3) ·7/12 + (2/2) ·5/18 = 21/36 + 10/36 = 31/36
Example 7d
• 2 1/3 + 5/9
• The curve ball here is the 2 in front. This
can show up in measurements! 2 1/3 inches
for example.
• We need to chop our 2 into 3rds so it can be
combined (added) to the 1/3 bit.
• 6/3 = 2 right?
• So the first term is 6/3+1/3 = 7/3 follow?
Example 7d finished
• So now we have 7/3+5/9 and we can keep going
as we have been finding a LCD (least common
denominator)
• Did you notice that the 3 in the bottom of the first
term already goes into 9 (the bottom of the
second term) evenly? We only have to work with
the first term.
• If we multiply the first term by 3/3 we get:
(3/3) ·7/3 +5/9 = 21/9 + 5/9 = 26/9
finis! **EX 73-84**
Cleaning up fractions with
decimals
• Practice fractions operations, especially
addition and subtraction (because they need
common denominators). Only practice will
make it CLICK in you!
• The last section here looks at special
fractions that have 10, 100, 1000 and the
like in the denominator
Fractions, Decimals, Percents
• If you have a calculator, you can check
these, but you can quickly see fractions and
percentages with denominators that are
multiples of 10
• 3/10 = 0.3, 8/10 = 0.8, 25/100 = 0.25,
• 10/1000 = 1/100 = .01 , 5/1000 = .005
• Play with these in the answer sheets also
Percentages
• If you take any of the numbers above and
multiply by 100, you get a percentage
• 3/10 = 0.3 [30%], 8/10 = 0.8 [80%],
25/100 = 0.25 [25%],
• 10/1000 = 1/100 = .01 [1%]
• 5/1000 = .005 [0.5%]
Example 8 (page 21)
• We want a denominator that is a multiple of 10
to make percentages easy to see.
• a) 1/5 if we multiply by 1 that is 20/20 then
we get (20/20)(1/5) = 20/100 =.20 =20%
• b) Going backwards 6% = .06 = 6/100 = 3/50
• c) 0.1 = 1/10 = (10/10)(1/10)=10/100 or 10%
**EX 85-96**
Example 9
• Since this is worked out well, I’ll leave it to
you to play with (all the steps are there).
• Hint: there are 3 boards 1 ½ inches wide, so
it starts with 3 ·1 ½
That’s where THAT comes from.
• AND it’s time to work with the exercises in
Section 1.2.
On to Section1.3
Adding/Subtracting REAL
numbers
• What is a Real number again?
• Any number that is on the number line.
• OR: How do we add and subtract numbers that
are both positive AND negative?
• From my experience, this will mess you up on
problems more than almost anything. Keep track
of minus signs!!
The easy part…
• If both numbers have the same sign, just add them (add
their absolute value) and keep the sign as it is in both.
EASY!
• (-10) + (-5) = -15
• Why do I put parenthesis around both numbers? Because
we don’t usually put +- next to each other. Historical and
for clarity. It would look like –10+-5 = -15
• We do sometimes just drop the + sign
• (-10) + (-5) = -10-5 = -15 (oh! That makes it easier).
Example 1 (page 27) Like Signs
• a) 23+56 =79
• b) (-12)+(-9)=-12-9= -21
• c) (-3.5)+(-6.28) = -3.5-6.28 = -9.78
add the absolute value of 3.5+6.28 then affix the • d) (-1/2)+(-1/4) = oops…
We need common denominators!
(2/2) (-1/2)+(-1/4)= (-2/4)+(-1/4)=-2/4-1/4=-3/4
**EX 7-16**
Unlike Signs
• If we add unlike signs, it is like adding a
debt to a savings.
• -10 + 10 = 0
• -5 + 10 = 5
• For ease, you can exchange the places of
things that are added. What does that mean?
Well… 5+3 = 8 and so does 3+5 =8 see?
The order doesn’t matter
•
•
•
•
-5 + 10 = 5
We could also write (10) + (-5) =5
Or even 10-5 =5
Wow, why didn’t we say so in the first
place?
The additive inverse property
• If you add a number to it’s negative, you get
zero every time. Duh.
• a + (-a) =0 and -a + a =0
• In numbers; 17 –17=0 and –17+17 =0
Example 2
(pg 28)
• a) 34+(-34) = 34-34 = 0
• b) -1/4+1/4 = 0
• c) 2.97+ (-2.97) = 2.97-2.97 = 0
**EX 17-20**
Watching the sign – Example 3 pg 29
• a) –5+13 = 13-5 = 8 (the sign of the answer
is the same as the number with the biggest
absolute value. 13 is further from 0 than 5)
• b) 6+(-7) = 6-7 = -1 (7 won this time and
was negative)
• c) -6.4 +2.1 = 2.1-6.4 = -4.3
• d) -5+0.09 = 0.09 – 5 = -4.91
Example 3 continued
• e) (-1/3)+(1/2) [note, ½ has a greater
absolute value, so the answer will be
positive]
We need common denominators again… 
(2/2)(-1/3)+(3/3)(1/2) = (-2/6)+(3/6) = 3/6-2/6
= 1/6 and it IS positive.
**EX 21-30**
Example 3f
•
•
•
•
•
3/8+(-5/6)
Sigh, LCD time.
Multiples of 8 = 8,16,24,32
Multiples of 6 = 6,12,18,24,30
Ok (3/3)(3/8) + (4/4)(-5/6) =
9/24+(-20/24)= 9/24-20/24 = -11/24
Example 4 (page 30)
• What if you have the negative of a
negative?
• What is - (-4) ? It is positive!
• a) –5-3= -8
• b) 5-(-3) = 5+3= 8
• c) –5-(-3) = -5+3= 3-5 = -2
• d) 1/2 – (-1/4) = (2/2)(1/2)-(-1/4)=2/4-(-1/4)
=2/4+1/4=3/4
Example 4 part 2
• e) –3.6-(-5) = -3.6+5= 5-3.6= 1.4
• f) 0.02 –8 = -7.98
**EX 31-58**
Let’s spend some time working on the hard parts of
section 1.3’s questions.
Section 1.4 – Real numbers and
multiplication and division!
• As we saw earlier, multiplication is often
easier than addition and subtraction
• Division is just the ‘upside down’ of
multiplication. Flip it and multiply if you
are asked to divide.
Notation
• So far, I’ve used the “·” symbol to show you
multiplication (most of the time)
• So multiplying a and b looks like a·b, but do we
really need a dot?
• People have decided to keep the + and – symbols
and drop the dot. a·b = ab
• And since division is the upside down of
multiplication I don’t need the ÷ on the keyboard
either. Doesn't that make life nicer? (We do use
the / for fractions, which is division).
What happens to the sign?
• The product is positive if the signs are alike.
•
a·a= positive answer
-a·-a = positive answer
• The product is negative if the signs are
unlike.
• -a ·a = negative answer a·-a = negative
answer
Example 1
pg 35
• a) (-2)(-3) = 6 [like signs - pos]
• b) 3(-6) = -18 [unlike signs - neg]
• c) -5 ·10 = -50 [unlike signs - neg]
• d) (-1/3)(-1/2) = 1/6 [like signs – pos]
• e) (-0.02)(0.08) = -0.0016 [unlike signs-neg]
• f) (-300)(-0.06) = 18 [like signs – pos]
**EX 7-18**
Division of Real Numbers
• The number on the bottom (the
denominator) can’t be 0 - this equals
infinity and is often given the symbol ∞
• so a divided by b = c, then b cannot be
equal to 0
Division: What sign?
• It’s the SAME rules as with multiplication
(since this is just upside-down
multiplication)
• If we have 10/(-2) it equals –5
• If we have –10/2 it equals –5
• If we have –10/-2 it equals 5
• If we have 10/2 it equals 5
Recapping signs
• If the signs in division (like multiplication)
are the same, the answer is positive
• If the signs in division (like multiplication)
are different, the answer is negative
• Another way to see this is like reducing
-5/-10 is like (-1/-1)(5/10) = (1)(5/10)= 1/2
Example 2 pg 36
• a) (-8) ÷ (-4) = (-1/-1)(8/4) = 1(2) =2 same
• b) -8 ÷ 8 = -1/1 = -1 different signs
• c) 8 ÷(-4) = -2 different signs
• d) -4 ÷ (1/3) tricky!
Lets invert (flip the 1/3rd upside-down) and
multiply -4(3/1)= -4(3)= -12 different
signs
Example 2 part 2
• e) –2.5 ÷ 0.05 use a calculator? Sure, but
you can do it with symbols!
-2.5 ÷ 0.05 we want to get rid of a decimal on
the top and bottom, what a mess!
(-2.5/0.05)(100/100) = -250/5 = -50
different signs = negative
Example 2 pt 3
• f) 0 / (-6) = 0/(-6) = 0
(note the zero is on top, not the bottom, so
nothing is blowing up to infinity)
**EX 19-32**
A few more notes
• In case you haven’t figured it out yet… the
÷ is the same as / symbols.
• 10 ÷ 3 IS the same as 10/3
(the dots are stand-ins for numbers)
• And back to zeros
8 ÷ 0, 8/0, 0 ÷0, 0/0 are all undefined (or
infinite …∞)
Playing with it!
• Run through some of the 1.4 exercises. Try
the ones under the topic that is hardest to
you. We’ll get to 1.5 shortly.
Section 1.5 Exponential
Expressions and the Order of
Operation
• What if the expression you need to work
with (starting with just numbers, but later
with letters or variables) that has addition,
subtraction, multiplication, division, and
exponents in it. Where to you start?
• It makes a difference!
How it matters…
• What if you have 3+2·5 what is the
answer?
• If you add first you get (3+2)·5 =5 ·5 =25
• If you multiply first you get
3+(2 ·5)= 3+10=13
WHICH IS RIGHT? 13 does not equal 25!!
Sometimes the writer helps with
grouping symbols
• EXAMPLE 1 page 40
• a) (3-6)(3+6) Do the things inside the
grouping symbols ‘( )’ first!
= -3 ·9 = -27
b) |3-4| - |5-9| Absolute value symbols can
group stuff as well! = |-1|-|-4| = 1-4 = -3
Example 1 part 2 pg 41
• c) The numerator (top) and denominator
(bottom) of a ratio or fraction need to be
done by themselves.
4-(-8)/5-9 = 4+8/5-9 = 12/(-4) = -3
**EX 7-18**
Exponential Expressions
• Here’s a new one for you!
• What if we multiply the same number to
itself many times. Why do we need to write
2 ·2 ·2 ·2 ·2 ·2 ? That takes up a lot of
space.
• Why not count how many 2’s there are and
code it for the next person … like this 26 ?
Example 2 pg 41
• a) 6 ·6 ·6 ·6 ·6 = 65
• b) (-3) (-3) (-3) (-3) = (-3)4
• c) 3/2 ·3/2 ·3/2 = (3/2)3
**EX 19-26**
Example 3 backwards
•
•
•
•
•
Going the other way…
y6 = y ·y · y ·y ·y ·y
(-2)4 = (-2)(-2)(-2)(-2)
(5/4)3 = (5/4)(5/4)(5/4)
(-0.1)2 = (-0.1)(-0.1)
pg 42
Example 4 getting the answer pg 42
• Evaluate these!
• 33 = 3 ·3 ·3 = 9 ·3 = 27
• (-2)3 = (-2)(-2)(-2) = 4(-2) = -8
• (2/3)4 = (2/3)(2/3)(2/3)(2/3)=4/9(2/3)(2/3)=
8/27(2/3) = 16/81
• (0.4)2 = (0.4)(0.4) = 0.16
**EX 33-46**
Caution
• As the book points out do NOT multiply the
number with the exponent, the exponent just
tells you how many of the number to
multiply together
• 33 = 3 ·3 ·3 = 9 ·3 = 27
• NOT 33 = 3 · 3 = 9
Another neat observation
• When dealing with negative numbers, you can
predict the sign of the answer just like we did
with like or unlike signs in multiplication and
division
• If you have an odd exponent, a negative number
is always negative answer
(-2)3 = (-2)(-2)(-2) = 4(-2) = -8 neg!
• If you have an even exponent, a negative number
is always a positive answer
(-2)4 = (-2)(-2)(-2)(-2) = -8(-2) = 16 pos!
Another Caution
• Make sure you know where the negative
sign is though!
• The above is true if you have the negative
sign INSIDE the parenthesis (-2)2
Not on the OUTSIDE –(2)2 =
- (2·2)= -(4) = -4
Example 5 playing with exponents
pg 43
• a) (-10)4 = (-10)(-10)(-10)(-10) = 10000 (positive
with even exponent)
• b) – 104 = -(10)(10)(10)(10) = -10000
• c) –(-0.5)2 = - (-0.5)(-0.5) = -(0.25) = -0.25
• d) –(5-8)2 = do the stuff in the ( )’s first
- (-3)2 = - (-3)(-3) = -9
**EX 49-56**
The Order of Operations
• So we don’t HAVE to put ‘( )’’s on
everything, we have a rule on how to
evaluate messy combinations of operations
like: 44 –5/6 +4 ·10
More to memorize
• 1. Do the things in groupings like ( ) and | |
first
• 2. Evaluate each exponent next (left to
right)
• 3. Do all the multiplications and divisions
next (left to right)
• 4. Do all addition and subtraction last
(from left to right)
How to remember them!
•
•
•
•
Please excuse my dear Aunt Sally
Parenthesis Exponents
Multiplication Division
Addition Subtraction
The left to right rule works even
for similar operations
• We know multiplication and division are
basically on the same level
• If we have 8 / 4 ·3 = 2 ·3 = 6
• Or for addition and subtraction (also on the
same level) - go left to right
• So 9-3+5 = 6+5 = 11
Example 6 pg 44
• 23 ·32 = do exponents first
= (2·2·2)·(3·3) = 8·9 = 72
• 2·5 -3·4+42 = 2·5 -3·4+16 (exponent first)
= 10-12+16 (do multiplication second)
= 14
Example 6 continued
2·3·4-33 + 8/2 = 2 ·3 ·4- (3)(3)(3) +8/2
= 2·3·4-27+8/2 the exponent goes first
Then multiplications and division left to right
= 6 ·4-27+4 = 24-27+4
Then the subtraction and addition left to right
= 24-23 = 1
**EX 57-72**
Example 7 more grouping and all
page 44
• a) 3-2(7-23) = 3-2(7-(2)(2)(2)) = 3-2(7-8)=
3-2(-1) = 3+2 = 5
• b) 3- |7-3·4| = 3- |7-12| = 3- |-5| = 3-5 = -2
Example 7 continued
• Is the same as :
(9-5+8)/(-52-3(-7))
since the top and bottom of a division (or fraction)
are already a group. We can put ( )’s around the and write them like this.
Exponent first (9-5+8)/(-25-3(-7) ) =
Then multiply/divide (9-5+8)/(-25+21)=
Then add/subtract 12/-4 =
The finish it -3
**EX 73-86**
If you have grouping inside
groupings Ex8 pg 45
• Start from the inside work and work your
way out
• a) 6- 4[5-(7-9)] = 6-4[5-(-2)] = 6-4[7] =
6-28 = -22
• b) -2|3- (9-5) |-|-3| = -2 |3-4| - |-3| = -2(1)-3
= -5
Section 1.5 Exercises
• Play with yet MORE of the worked
problems!!!
• Force yourself to work on the harder
concepts and get group help, answer sheet
help and my help!
1.6 Getting to the the Algebra!
• Now we’ll start using all the above basic games
one plays with numbers and stick in a letter for
one or more numbers. This letter means that
later you can enter numbers when you have
them.
• You don’t always get a numerical answer. You
stop with letters still in your answer
• This is more common in science, mathematics,
economics, accounting and the like. Use a
general equation and stick in the numbers later.
What is an algebraic expression?
•
•
•
•
•
•
x+2
πr2
b2-4ac
a-b/c-d
E=mc2
2πr
An expression by any other name
smells as sweet…
• We name them (just history again) by the last
operation one uses when plugging in all the
numbers and working them out
• x+2 = a sum
• a-bc = a difference
• 3(x-4) = product (you do the x-4 first)
• 3/(x-4) = quotient (you do the x-4 first again)
• (a+b)2 = is called a square (you do the a+b first)
Example 1 naming pg 49
• 3(x-2) = a product (x-2) is done first
• b2-4ac = a difference (you do the b2 and 4ac
first)
• a-b/c-d = a quotient (you do the top and
bottoms first)
• (a-b)2 = a square (you do the a-b first)
**EX 7-18**
Speaking the Talk
• On page 50 is a great guide to reading
equations out loud. The ‘verb’ of the
phrase is based on the name of the
expression from above (the last operation
done).
• Practice this…
Example 2 talking the talk pg 50
•
•
•
•
•
a)
b)
c)
d)
e)
3/x = the quotient of 3 and x
2y+1 = the sum of 2y and 1
3x-2 = the difference of 3x and 2
(a-b)(a+b) = the product of a-b and a+b
(a+b)2 = the square of the sum of a+b
**EX 19-28**
Example 3 pg 50 writing the write
•
•
•
•
a) The quotient of a+b and 5 = (a+b)/5
b) The difference of x2 and y2 = x2-y2
c) The product of π and r2 = πr2
d) The square of the difference x-y = (x-y)2
**EX 29-44**
Plugging in the numbers already
• Hey! We just switched to letters, now we’re
back to numbers. What’s up?
Anyway…
Evaluating Algebraic Expressions
• Plug in what you are given and you’re back
to numbers…
• What is the value of x-2y if x=-2 and y=-3
-2 –2(-3) = -2+6 = 4
Example 4 Pluggin’ in numbers pg 51
• You are given that a=3, b=-2, c= -4
• a) a2+2ab+b2 = 32+2(3)(-2)+(-2)2 =
Exponents first: 9+2(3)(-2)+ 4 =
Multiplication next: 9+ (-12)+4 =
Then addition/subtraction: = 13-12 = 1
More from example 4
• You are given that a=3, b=-2, c= -4
• b) (a-b)(a+b) = (3-(-2))(3+(-2)) =
Do inner groupings first (3+2)(3-2) =
Then outer groupings next (5)(1) = 5
Part 4c
•
•
•
•
You are given that a=3, b=-2, c= -4
b2-4ac
(-2)2 – 4(3)(-4)
4- 12(-4) = 4-(-48) = 4+48 = 52
Problem 4d
•
•
•
•
•
You are given that a=3, b=-2, c= -4
(-a2-b2)/(c-b)
(-(3)2-(-2)2)/(-4-(-2))
Exponents first (-9-4)/(-4-(-2))
Then inner groupings (-9-4)/(-4+2) =
(-13)/(-2) = 13/2
**EX 45-68**
What is an Equation?
• It is anything where one side equals the
other side of the = sign.
• Easy?
Examples…
•
•
•
•
•
•
11-5 = 6
100-4 = 96
x+3=9
2x+5=13
x/2-4 =1
For these last three, there is one number you
can stick into x that makes this true (it
satisfies it). That number is called the
solution or root. Find that number and you
have solved the equation.
A quick definition
• When testing an equation, we sometimes
put in a ? to show we don’t know if the
sides are equal.
• So x+1 =? 2
is true if x=1 but not if it
equals anything else.
Example 5 Are these the
answers? Page 52
• a) 6, 3x-7=9 plug 6 into the x
3(6)-7 = 18-7 = 11 =? 9 no!
• b) -3, (2x-4)/5=-2 plug –3 into the
(2(-3)-4)/5 = (-6-4)/5 = -10/5 =-2 =? –2 yes!
Example 5 continued
• c) -5, -x-2 = 3(x+6)
Oh oh, we have 2 sides to compare… not a
problem. Just plug in the number for all x’s
-(-5)-2 =? 3((-5)+6)
5-2 =? 3(6-5)
3=? 3(1)
3=?3 YES!!
**EX 69-82**
Example 6 writing the talk again
page 52
• Lost in translation?
• The sum of x and 7 is 12 : x+7=12
• The product of 4 and x is the sum of y and 5:
4x=y+5
• The quotient of x plus 3 and 5 is –1:
(x+3)/5 = -1
**EX 83-90**
Using an algebraic expression
• What if you cared not only for the solution of
the equation (what x equals) but for ALL
values of x?
• Example 7 shows an algebraic expression that
forensic scientists use. If you plug in MANY
values for F in 69.1+2.2F and plot the results,
you can get a graph that allows you to predict
things you’ve never measured!
More Exercises! Section 1.6
• SECTION 1.6 Problems. You have them all
worked out, focus on the tricky parts.
Section 1.7 Properties of Real
Numbers – Playing with them!
• We’ve seen this before, but now we are
using our algebraic equations (letters in for
some of the numbers).
• Put the the commutative property into your
vocabulary (you commute to work… you
move around, so you need to remember
“moving”)
The Commutative Property
• For any real numbers a and b:
a+b = b+a you can move them around for
addition and subtraction (so a-b=-b+a)
• AND
• ab = ba or 2(5) = 5(2)  10=10
Example 1 commuting without
moving violations page 58
• Rewrite these expressions using the
commutative properties
• a) 2 + (-10) = -10 +2
• b) 8+ x2 = x2 +8
• c) 2y – 4x = -4x +2y
**EX 7-12**
Example 2 and with
multiplication page 58
• a) n ·3 = 3 ·n = 3n
• b) (x+2) ·3 = 3(x+2)
• c) 5-yx = 5-xy we switched the yx
this could also read -xy+5 commute away!
**EX 13-18**
Associative Properties
• Things added or multiplied to each other
will associate (like mingling around at a
party… “let’s associate darling”).
The Associative Properties
• For real numbers a,b, and c
(a+b)+c=a+(b+c) and
(ab)c = a(bc)
It doesn’t matter how you group things just
added to one another, or just multiplied to
each other.
Example 3 Associating with the
right variables page 59
• a) (3x)(x) = 3x ·x = 3x2
• b) (xy)(5yx) = xy5yx = 5xxyy = 5x2y2
**EX 19-24**
Example 4 Working with
numbers page 60
• We can move positive and negative
numbers around and group them for ease
(using the associative property) …
• a) 3-7+9-5 = (3+9) + (-7-5) = 12-12 = 0
• b) 4-5-9+6-2+4-8 = (4+6+4)+(-5-9-2-8) =
14-24 = -10
• **EX 25-32**
Danger – Subtraction & Division
• Subtraction and Division are NOT
associative operations
• Example : (8-4)-3 = 4-3 = 1
8-(4-3) = 8-1 = 7
we only moved the ( )’s!!! Yikes!
• (16/4)/2 = 4/2 = 2
16/(4/2) = 16/2 = 8 Yikes again!
Again, we only moved the ( )’s!!!
The Distributive Property
• I LOVE this… very fun!
• You get to multiply into and between terms
in ( )’s.
• See how it works:
3(4+5) = 3(9) = 27
or 3(4) + 3(5) = 12 + 15 = 27 SAME!
The 3 is distributed into the sum!
Officially, the Distributive
Property
• a(b+c) = ab+ac
• a(b-c) = ab- ac
• You get to bust apart the grouping!
Example 5 distributing page 61
• a) a(3-b) = a3 – ab = 3a –ab
distributive commuting
• b) -3(x-2) = x(-3) –2(-3) = -3x +6
**EX 33-44**
Example 6 distributing
backwards page 62
• These can be a bit of a brain teaser, but fun also!
• a) 7x-21 notice you can see a 7 in both the first
and second number
= 7x-(7·3) = 7(x-3)
• b) 5a+5 = 5(a+1)
**EX 45-56**
A few more definitions…
• The identity property. We have already
noticed that 0 and 1 are special.
• 1 doesn’t change things much when
multiplication or division happens
• 0 kills anything it touches with
multiplication or division ,but doesn’t
matter in addition or subtraction
Identity Properties
• For any real number…
a ·1 = 1 ·a = a
And
a+0 = 0+a = a
And the Inverse Properties
• For any real number a there is another real
number –a that kills it.
a+(-a) =0
• a · 1/a = 1 (another way to look at it is the
a’s cancel).
examples
• If you start with 2/3 and multiply the
inverse (flip it) you get…
2/3 · 3/2 = 6/6 = 1
Or with 5 you get…
5/1 · 1/5 = 5/5 =1
Example 7 page 63
Multiplicative inverses
•
•
•
•
•
Find the multiplicative inverse of each:
a) 5 is 1/5
b) 0.3 = 3/10 so it’s 10/3
c) -3/4 is -4/3
d) 1.7 = 1(7/10) = 10/10 + 7/10 = 17/10 so it’s
10/17
**EX 57-68**
What about zero?
• It kills all.
• The multiplication property of zero
0 ·a = 0
and
a ·0 = 0
Example 8 name that process
page 63
• This example just names what is going on
and is a good review. I won’t cover it
through the power point notes.
• Look it over before working with the
problems.
• **EX 69-88**
Example – What good is an
inverse?
• You can see by this example that taking the
inverse turns a rate (cars/hour) or
(miles/hour) into a time per unit (hours/car)
or (hours/mile).
• I can make 10 power point slides an hour.
How long will it take to make 1 slide?
• 1/10 hours/slide = .1 hours/slide
Section 1.7 Problems
• For the 7th out of 8 times tonight, we’ll
pause for problems at the end.
• Remember, you have the solutions to go
over every one of these eventually.
• If you really want the SKILL of math in
your brain, you have to practice it!
Section 1.8 Putting it all together
to solve, simplify and compute
• You can use all the tools you have now to
work with numbers and algebraic
expressions.
• Let’s do it!
Can you do this without help?
• If you have (26)(200), can you do this in
your head?
• What if you move things around a bit
(associative property)?
• (26)(2 ·100) = (26 ·2)(100)=
52 ·100=5200
Example 1 pg 67
Grouping for simplicity
• a) 347 +35 + 65  note 35+65 =100! 
347 +(35+65)=347+100=447
• b) 3 ·435 ·1/3  note it’s easy to combine
the 3 and 1/3!  (3 ·1/3) ·135=
1 ·135=135
More on Example 1
• c) 6 ·28+4 ·28  note that both sides of
the + have a 28 in them 
(6+4)28 = 28(6+4) = 28(10) = 280
**EX 7-22**
Like Terms
• Another definition for you (when your head
is about to explode) = Like Terms
• Like terms are terms where the variables are
the same letters and raised to the same
powers. You can combine this with addition
and subtraction. The coefficients (the
number before the terms) add and subtract.
Examples
•
•
•
•
-3 and 6 and 10 and –4 can all combine
5x and 14x and 10x and –4x can all combine
6x2 and x2 and –4x2 and 1004x2 can all combine
abx3 and –51abx2 and 1/2abx2 can all combine
• See the pattern here?
Example 2 – combine them there
terms! Page 67
• a) 3x+5x = (3+5)x = 8x
• b) -5xy – (-4xy) = (-5-(-4))xy = (-5+4)xy
= -xy
• Bonus: 5x+14y+2x-4y
= (5x+2x)+ (14y-4y) = (5+2)x+(14-4)y=
7x+10y
**EX 23-28**
Example 3 – going quicker page 68
•
•
•
•
•
a) w+2w = 3w
b) –3a+(-7a) = -10a
c) –9x+5x = -4x
d) 7xy –(-12xy) =19xy
e) 2x2+4x2 = 6x2
**EX 29-42**
Caution…
• If no terms are like then you can’t do
anything… for example…
• 2+5x
• 3xy+5y
• 3w+5a
• 3z2+5z
• 100+4xz+20y3
Products and Quotients
(multiplication and division)
• We can use the associative property of
multiplication to simplify the product of two
expressions… let’s do it!
• (Showing is often better than telling)
Example 4 ; Products page 69
•
•
•
•
Simplify these…
a) 3(5x) = (3 · 5)x = 15x
b) 2(x/2) = 2/1(x/2) = 2x/2 = 1 · x = x
c) (4x)(6x)  remember it’s all multiplied
so we can commute  4 ·6 ·x ·x = 24x2
• d) (-2a)(4b)  doing the same trick 
-2 ·4 ·a ·b = -8ab
**EX 43-52**
Example 5 going quicker
•
•
•
•
page 69
a) (-3)(4x) = -12x
b) (-4a)(-7a) = 28a2
c) (-3a)(b/3)  (three’s cancel)  -ab
d) 6 · x/2 = 6/2(x) = 3x
**EX 53-58**
Example 6 now the same for
quotients page 70
• Simplify
• a)10x/5 = 1/5 ·10 ·x = (1/5 ·10) ·x= (2)x =2x
• b) (4x+8)/2 = ½(4x+8) = (4/2)x+ (8/2) = 2x+4
**EX 59-70**
Removing Parentheses
• If you multiply a grouping by –1, you can
make things look simpler easily…
• (-1)(7) = -7
• (-1)(-8) = 8
• (-1)(x) = -x
• -1(y+5) = -(y+5) = -y –5
• -(x-3) = -1(x-3) = -1x – (-3) = -x+3
• Just take it step by step and all will be good!
Example 7 page 71
• Removing Parentheses – simplifying
• a) -(x-4)+5x-1 = -x+4+5x-1 = -x+5x+4-1 = 4x+3
• b) -(-5-y)+2y-6 = 5+y+2y-6 = 3y-1
• c) 10-(x+3) = 10-x-3 = -x+7
• d) 3x-6-(2x-4) = 3x-6-2x+4 = x-2
**EX 71-86**
Putting everything together
• Now we can simplify anything by
multiplying out terms, combining term,
moving terms around (distributing,
associating, commuting).
Example 8 page 71
Simplifying algebraic equations
• a) (-2x+3) + (5x-7) = -2x +3 +5x –7 =
-2x+5x +3-7 = -3x – 4
• b) (-3x+6x)+5(4-2x) = -3x+6x+20-10x =
-3x+6x-10x +20 = -7x+20
Example 8 continued
• c) -2x(3x-7) – (x-6) =
-6x2+14x -x +6 = -6x2 +13x +6
• d) x-0.02(x+500) =
x- 0.02x + (-0.02)(500) = x-0.02x- 10=
0.98x –10
**EX 87-104**
And finally – play with the 1.8
exercises
• You again have the answers to those
problems not assigned
• Practice is SOOO important in this course.
• Do everything you can scrape time up for,
first the hardest topics then the easiest.
• You are building a skill like typing, skiing,
playing a game, solving puzzles.
Quiz on ALEKS
• Right now due Tuesday night at 10pm.
• If we didn’t get far enough, I’ll move the
quiz off to next week.