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College Algebra I
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 1,2,3,4,8 for this part, part I).
• I’ll supply the worked answers for ALL non-assigned problems
• 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 (pg3) – 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.
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.
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 Chapter 9
Page 5 ‘who owns who’ in the sets we’ve covered…
• Who belongs as part of who in the number definition game?
See the squares on page 5.
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.
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 4 (pg 6)
• These examples are having you practice dropping negative signs off numbers.
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 5 (pg 7)
• 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.
• Let’s go over the Warm-ups as well.
End of 1.1 Exercises
• I’m passing out the answers to all non-assigned problems.
• 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 car 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, 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 a/a = 1
• If you multiply ANYTHING by 1 it is the same as it was before
• 81 = 8 ½ 1 = ½ etc.
• 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 9
Example 1 pg 9
• To solve these, you just want to know what form of 1 was secretly used to get the 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.
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 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
• 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.
Which takes 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 = (11)/(23)= 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
• 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!
Example 4• 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!
Example 4 b
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 of Example 4b
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 2/13/11 = (23)/(111)=6/11
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/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 12 (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)=(12)/(17)= 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!)
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 ate2/6+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 12
• 1/7 + 2/7 = (1+2)/7 = 3/7
• 7/10 - 3/10 = (7-3)/10 = 4/10 = (22)/(25)= (22)/(25)= 2/5
reducing by expanding so we can see the 2/2
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?
• 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 7
• 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=
(31)/(32)+(21)/(23) = 3/6+2/6=5/6We 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• 3/4-1/6 • 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
4,8,12,16,20,24 etc…• And the second 6,12,18,24 etc…• We could use 12 or 24. Both numbers have that
multiple in common. 12 is the lowest (most reduced)• So we will multiply both fractions by something equal
to 1 that makes the bottoms of both equal to 12
(3/3) 3/4 – (2/2) 1/6 = 9/12 – 6/12 = 3/12
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!
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
• 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%
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. I’m passing out the answers to the non-assigned problems again.
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?
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 Like Signs
• a) 23+56 =79
• b) (-12)+(-9)=-12-9= -21
• c) (-3.5)+(-6.28) = -3.5-6.28 = -9.78
• 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
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
• a) 34+(-34) = 34-34 = 0
• b) -1/4+1/4 = 0
• c) 2.97+ (-2.97) = 2.97-2.97 = 0
Watching the sign – Example 3 pg 21
• 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.
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
• 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
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 ab, but do we really need a dot?
• People have decided to keep the + and – symbols and drop the dot. ab = 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.• aa= 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 25
• 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]
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
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 26
• 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)
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
• 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+25 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?
Sometimes the writer helps with grouping symbols
• EXAMPLE 1
• 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 suff as well! = |-1|-|-4| = 1-4 = -3
Example 1 part 2 pg 29
• 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
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 30
• a) 6 6 6 6 6 = 65
• b) (-3) (-3) (-3) (-3) = (-3)4
• c) 3/2 3/2 3/2 = (3/2)3
Example 3 backwards pg 31
• 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)
Example 4 getting the answer
• 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
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
• 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 = - (22)= -(4) = -4
Example 5 playing with exponents
• 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
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 33
• 23 32 = do exponents first = (222)(33) = 89 = 72
• 25 -34+42 = 25 -34+16 (exponent first)
= 10-12+16 (do multiplication second)
= 14
Example 6 continued
234-33 + 8/2 = 2 3 4- (3)(3)(3) +8/2
= 234-27+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
Example 7 more grouping and all
• a) 3-2(7-23) = 3-2(7-(2)(2)(2)) = 3-2(7-8)= 3-2(-1) = 3+2 = 5
• b) 3- |7-34| = 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
If you have grouping inside groupings
• 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
• 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)
Speaking the Talk
• On page 37 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
• a) 3/x = the quotient of 3 and x
• b) 2y+1 = the sum of 2y and 1
• c) 3x-2 = the difference of 3x and 2
• d) (a-b)(a+b) = the product of a-b and a+b
• e) (a+b)2 = the square of the sum of a+b
Example 3 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
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 38
• 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
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?
• 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!!
Example 6 writing the talk again
• 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
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 foensic 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
• Rewrite these expressions using the commutative properties
• a) 2 + (-10) = -10 +2
• b) 8+ x2 = x2 +8
• c) 2y – 4x = -4x +2y
Example 2 and with multiplication
• 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!
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
• a) (3x)(x) = 3x x = 3x2
• b) (xy)(5yx) = xy5yx = 5xxyy = 5x2y2
Example 4 Working with numbers
• 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
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) = 27or 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
• a) a(3-b) = a3 – ab = 3a –ab
distributive commuting
• b) -3(x-2) = x(-3) –2(-3) = -3x +6
Example 6 distributing backwards
• 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-(73) = 7(x-3)
• b) 5a+5 = 5(a+1)
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 7Multiplicative 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
What about zero?
• It kills all.
• The multiplication property of zero
0 a = 0 and a 0 = 0
Example 8 name that process
• 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.
Example 9 – 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 52Grouping 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 pg 52
• 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
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 cam all combine
• See the pattern here?
Example 2 – combine them there terms!
• 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
Example 3 – going quicker
• a) w+2w = 3w
• b) –3a+(-7a) = -10a
• c) –9x+5x = -4x
• d) 7xy –(-12xy) =19xy
• e) 2x2+4x2 = 6x2
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
• 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
Example 5 going quicker
• a) (-3)(4x) = -12x
• b) (-4a)(-7a) = 28a2
• c) (-3a)(b/3) (three’s cancel) -ab
• d) 6 x/2 = 6/2(x) = 3x
Example 6 now the same for quotients
• 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
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
• Removing Parentheses – simplifying
• a) 5-(x+3) = 5 –x –3 = -x +5 –3 = -x +2 = 2-x
• b) 3x-6-(2x-4) = 3x-6-2x-(-4) = 3x-6-2x+4= 3x-2x-6+4 = x-2
• c)-6x-(-x+2) = -6x-(-x)-2 = -6x+x-2=-5x-2
Putting everything together
• Now we can simplify anything by multiplying out terms, combining term, moving terms around (distributing, associating, commuting).
Example 8Simplifying 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
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.
