Chapter 3
Group 1
3.1
3.1.1
Order of operations
Order of operations with numbers
Example 1. Simplify the following:
−(3 − 5) − [2 − (32 − 13)]
(Here’s a fun video explaining some of the order of operations, and the reasons behind them: http://www.youtube.
com/watch?v=y9h1oqv21Vs. Note: the video is so fast, you’ll need to pause it!)
Solution.
−(3 − 5) − [2 − (32 − 13)] = −(−2) − [2 − (9 − 13)]
= 2 − [2 − (−4)]
= 2 − (2 + 4)
= 2−6
= −4
Example 2. Simplify the following:
√
62 − 3 25
62 + 13
Solution. Note: you can’t cancel the 62 on the top and bottom like this
√
2 − 3 25
6�
�
Wrong:
2 + 13
6�
�
The right way to look at a fraction like this is that there are implied parentheses on the top and bottom
√
(62 − 3 25)
(62 + 13)
It’s like the numbers on top are married to each other, and the same with the numbers on the bottom. By trying to
cancel the 62 , we’re breaking up the marriage.
The right way to do it is to work inside the parentheses.
21
CHAPTER 3. GROUP 1
22
√
(62 − 3 25) (36 − 3 · 5)
=
(62 + 13)
(36 + 13)
(36 − 15)
=
49
21
=
49
3
=
7
3.1.2
Order of operations with algebra
Example 3. Simplify the following:
(3y3 + 9y2 − 11y + 8) − (−4y2 + 10y − 6)
Solution.
(3y3 + 9y2 − 11y + 8) − (−4y2 + 10y − 6) = 3y3 + 9y2 − 11y + 8 − (−4y2 ) − 10y − (−6)
3y3 + 9y2 − (−4y2 ) − 11y − 10y + 8 − (−6)
3y3 + 9y2 + 4y2 − 21y + 8 + 6
3y3 + 13y2 − 21y + 14
Example 4. Simplify the following:
(3x − 1)(x + 2) − (2x + 5)2
Solution. The main step is FOIL: (a + b) (c + d)
(3x − 1)(x + 2) − (2x + 5)2 = [3x2 + 6x − x − 2] − [4x2 + 20x + 25]
= 3x2 − 4x2 + 6x − 20x − 2 − 25
= −x2 − 14x − 27
3.2
3.2.1
Linear Equations
Graphing linear equations
Example 5.
(a) Make an assumption about how many books, on average, that you buy a month: # books/month =
.
(b) Make an assumption about how much each book costs, on average, if you buy it in paper: Avg cost paper =
.
(c) Make an assumption about how much each book costs, on average, if you buy it as an ebook: Avg cost ebook =
.
(d) Make an assumption about how much an ebook reader costs: ebook reader cost =
CHAPTER 3. GROUP 1
23
(e) Plot/graph a set of points showing how much you’ll spend, total, after buying paper books. Plot one point for
the total you’ve spent after 1 month, another point for the total after 2 months, a third point for the total after 3
months, etc. Make 30 points, one for each month from 1 to 30. (You can use the graph paper on the next page,
you’ll need to choose an appropriate scale for the vertical axis). Draw a straight line through all the points.
(f) On the same graph as the previous part, draw another set of points showing how much you’ll spend, total, after
buying ebooks for t months.
(g) On a purely economic level, looking at the next 30 months, should you buy an ebook reader?
Solution.
(a) I’ll assume that I buy on average 2 books/month.
(b) I’ll assume that each paper book costs $15.
(c) I’ll assume that each ebook cost $10.
(d) I’ll assume that an ebook reader is $100.
(e) At the end of one month of paper books, I’ll spend 2 · $15. At the end of two months I’ll have spent a total of
$30 + $30 = 2 · 30. After three months it will be 3 · $30, etc. If I plot 30 moths worth of ebooks and 30 months
worth of paper books at the same time, I get this:
The circles are the ebook cost and the crosses are the paper book totals. It looks after 10 months the total cost
for ebooks will cost less. Thus, on a purely economic basis, the ebook is better.
Example 6. Suppose during a 5-year period the profit P(Y ) (in billions of dollars) for a large corporation was given
by P(Y ) = 7 + 2Y , where Y represents the year.
CHAPTER 3. GROUP 1
(a) Fill in the chart.
24
Y
0
P(Y )
1
2
3
4
(b) What are the units of P(Y )?
(c) What does the 2 in the equation represent, and what are the units?
(d) What was the initial profit?
Solution.
(a)
Y
0
P(Y ) 7
1
9
2
11
3
13
4
15
(b) The function gives profit, in billions of dollars, so the units are $ Billion.
(c) The 2 represents how much the profit increases per year.
(d) The initial profit is the profit in year 0: 7 billion dollars.
3.2.2
linear equations
Example 7. Most lines can be written with an equation like this y = mx + b, where b is the y-intercept and m is the
slope of the line. The y-intercept is where the line hits the y-axis, and the slope is a ratio of how much the line rises,
divided by how much far the line goes horizontally. This form of a line’s equation is the called slope-intercept form.
Plot the following lines (there are four sets of graph paper on the next page).
(a) y = 3x + 2
1
(b) y = − x − 5
2
2
(c) y = − x + 5
3
(d) y = 5x − 7
Example 8. There are other forms we can write the equation of a line in. Some of them are more convenient to write,
but they can usually be turned into slope-intercept anyway.
Turn the following equations into slope-intercept:
(a) 3y + 2x = 5
(b) −y + 3x = 6
(c) y − 5 = 3(x − 2)
(d) y = −4(x + 2) + 10
Solution.
(a)
3y + 2x = 5
3y = −2x + 5
2 5
y=− +
3 3
CHAPTER 3. GROUP 1
25
(a)
(b)
(c)
(d)
(b)
−y + 3x = 6
−y = −3x + 6
y = 3x − 6
(c)
y − 5 = 3(x − 2)
y = 3(x − 2) + 5
y = 3x − 6 + 5
y = 3x − 1
CHAPTER 3. GROUP 1
(d)
y = −4(x + 2) + 10
y = −4x − 8 + 10
y = −4x + 2
26
CHAPTER 3. GROUP 1
27
Example 9. To write the equation of a line through two points you need to some information: (1) a slope and yintercept, or (2) a slope and a point, or (3) two points.
If you have are given the slope m and a point (x0 , y0 ), the easiest way to write the equation of a line is to use the
point-slope form:
y = m(x − x0 ) + y0
where you are given m and (x0 , y0 ).
(a) Find the equation of a line through the point (−2, 3) with slope 5. Start with the point-slope form, and then put
it into slope-intercept form.
(b) Find the equation of a line through the point (−2, 3) with slope −3/4. Start with the point-slope form, and then
put it into slope-intercept form.
Solution.
(a)
y = m(x − x0 ) + y0
y = 5(x + 2) + 3
y = 5x + 10 + 13
y = 5x + 23
(b)
y = m(x − x0 ) + y0
3
y = − (x + 2) + 3
4
3
3
y = − x− ·2+3
4
4
3 3
y = − − +3
4 2
3
3
y = − x+
4
2
Example 10. The fastest way to find the equation of the line through two points is to (1) calculate the slope, and then
(2) use the point-slope form of a linear equation.
(a) Find the equation of the line through the points (1, 4) and (3, 7).
(b) Find the equation of the line pictured below:
CHAPTER 3. GROUP 1
Solution.
28
(a) Find the equation of the line through the points (1, 4) and (3, 7).
rise
run
7−4 3
m=
=
3−1 2
y = m(x − x0 ) + y0
3
y = (x − 1) + 4
2
m=
(b) It appears that the line goes through (3, 0) and (0, 3.5).
rise
run
3.5 7
m=
=
3
6
y = m(x − x0 ) + y0
7
y = x + 3.5
6
m=
Example 11. The fastest way to match lots of graphs of lines with lots of equations of lines, not necessarily written
in slope-intercept form, is to find the x and y intercepts. To find these, simply plug in y = 0 or x = 0 and then solve for
the other variable.
Match the following equations with the correct graph.
(a) 2x + 2y = 4
(b) 2x − 2y = 6
(c) −3x − 3y = 9
(d) −2x + 3y = −12
A
B
CHAPTER 3. GROUP 1
C
29
D
Solution. Note: the way I thought this problem should be done was to find the x and y intercepts. But many of you
simply rewrote the equation of the line in slope-intercept form, and maybe your’re right, maybe that’s just as fast.
To find the y-intercept, plug in x = 0 and solve for y. To find the x-intercept, plug in y = 0 and solve for x.
(a)
2x + 2y = 4
x = 0 ⇒ 2y = 4 ⇒ y = 2
y = 0 ⇒ 2x = 4 ⇒ x = 2
This matches the graph A.
(b)
2x − 2y = 6
x = 0 ⇒ −2y = 6 ⇒ y = −3
y = 0 ⇒ 2x = 6 ⇒ x = 3
This matches the graph B.
(c)
−3x − 3y = 9
x = 0 ⇒ −3y = 9 ⇒ y = −3
y = 0 ⇒ −3x = 9 ⇒ x = −3
This matches the graph D.
(d)
−2x + 3y = −12
x = 0 ⇒ 3y = −12 ⇒ y = −4
y = 0 ⇒ −2x = −12 ⇒ x = 6
This matches graph C.
CHAPTER 3. GROUP 1
3.3
3.3.1
30
Fractions
Fractions with numbers
1
1
Example 12. The simplest fractions are ones like or . You have one thing (a line, a cookie, a dollar,etc.) and you
2
5
are going to divide it among 2 or 5 people.
One cookie divided by two people gives the same amount of cookie as 4 cookies divided by 8 people: we each get
half and so
1 4
=
2 8
Similarly, two dollars divided by 10 people is the same as 3 dollars divided by 15 people because
3
2
1
=
=
15 10 10
There are two ways you can check if two fractions are equal: (1) You can reduce them both and see if you get the same
thing, (2) you can cross multiply and see if the results are equal:
7
35
7
5·7
7
✁
5·7
✁
1
5
(a) Reduce the fraction:
55
.
99
(b) Reduce the fraction:
−35
.
55
9
45
? 9
=
5·9
9
? ✁
=
5 · 9✁
? 1
= �
5
?
=
OR
7 ? 9
=
35 45
?
7 · 45 = 9 · 35
?
315 = 315�
(c) Reduce both fractions to see if these fractions are equal:
42 ? 48
= .
60 88
(d) Reduce both fractions to see if these fractions are equal:
90 ? 75
=
.
120 100
(e) Cross multiply to see if these fractions are equal:
87 ? 93
=
.
116 124
(f) Cross multiply to see if these fractions are equal:
156 ? 234
=
.
228 342
(g) Cross multiply to see if these fractions are equal:
527 ? 703
=
.
899 1147
Solution.
(b)
(a)
55 5 · 11 5
=
= .
99 9 · 11 9
−35 −5 · 7
7
=
= .
55
5 · 11
11
CHAPTER 3. GROUP 1
(c)
(d)
31
42
6·7
7
=
= .
60 6 · 10 10
48
6·8
6
=
= .
88 8 · 11 11
These are not equal.
90
9 · 10
9
3·3 3
=
=
=
= .
120 12 · 10 12 3 · 4 4
3 · 25 3
75
=
= .
100 4 · 25 4
These are equal.
(e)
87 ? 93
=
116 124
?
87 · 124 = 93 · 116
?
10788 = 10788�
They are equal.
(f)
156 ? 234
=
228 342
?
156 · 342 = 234 · 228
?
53352 = 53352
They are equal.
(g)
527 ? 703
=
899 1147
?
527 · 1147 = 703 · 899
?
604469 = 631997
They are not equal.
2
1
1
Example 13. By definition a fraction like means that you have “two fifths,” i.e. one and another . Thus,
5
5
5
� �
2 1 1
1
2
= + = 2·
and the expression answers the question “how many fifths do I have? I have two.”
5 5 5
5
5
2 1
This is why we can add fractions that have the same denominator, the same stuff on the bottom: + means “I
5 5
have some fifths, in fact two of them. You have some fifths, in fact one of them. How much do we have all together?
My two plus your one is three, a total of three fifths:
2 1 3
+ =
5 5 5
Subtraction works the same way.
CHAPTER 3. GROUP 1
32
(a) Add the fractions, and simplify if possible:
3
2
+ .
10 10
(b) Add the fractions, and simplify if possible:
5
7
+ .
14 14
(c) Subtract the fractions, and simplify if possible:
11
5
−
12 12
(d) Subtract the fractions, and simplify if possible:
19 22
− .
30 30
(e) Add the fractions, and simplify if possible:
17 3
+ .
x
x
(f) Add the fractions, and simplify if possible:
3x 5x
+ .
16 16
Solution.
(a)
2
5
1
3
+
=
= .
10 10 10 2
(b)
5
7
22 11
+
=
= .
14 14 14
7
(c)
11
5
6
1
−
=
= .
12 12 12 2
(d)
19 22 −3
1
−
=
=− .
30 30
30
10
(e)
17 3 17 + 3 20
+ =
= .
x
x
x
x
(f)
3x 5x 3x + 5x 8x
x
+
=
=
= .
16 16
16
16 2
Example 14. What happens when you try to add different kinds of things? Can I combine 2x + 3y and get 5(x + y)?
(No!) If you have 2 books and I have 3 chairs, do we have 5 “book+chairs”? (No!) Well, in some sense, but I don’t
know what a “book+chair” is, and I know that I don’t have 5 books and I don’t have 5 chairs.
2 3
The formula + can be read as “I have some fifths of a cookie, in fact I have two of them. You have some
5 4
fourths of a cookie. In fact, you have three of them. How many cookies do we have all together?” Well, just like we
can’t directly add books and chairs, we can’t directly add fifths and fourth. A fifth plus a fourth is not a fifth+third.
The above discussion explains why you need a common denominator to add fractions. We can’t directly combine
a fifth and a fourth, we need to rewrite them as something else.
We say that two fractions have a common denominator when we find a way to rewrite one or both of the fractions
so that the same number appears on the bottom. Once we do this, then we can write both fractions with a common
denominator, and add them like we did before.
In the following problems, reduce your answer. If the answer is “improper” (i.e. the top is bigger than the bottom)
leave it that way. Don’t rewrite it as a “mixed” fraction (i.e. something like 11/2).
(a) Get a common denominator and combine the fractions:
1 3
+ .
2 4
(b) Get a common denominator and combine the fractions:
5
2
− .
6 15
CHAPTER 3. GROUP 1
33
(c) Get a common denominator and combine the fractions:
3
8
+ .
10 15
(d) Get a common denominator and combine the fractions:
3
2
+ .
7 11
(e) Get a common denominator and combine the fractions:
3
x
+ .
7 11
(f) Get a common denominator and combine the fractions:
3 2
+ .
7 x
Solution.
(a)
1 3 2 3 5
+ = + = .
2 4 4 4 4
(b)
5
2
25
4
21
7
−
=
−
=
= .
6 15 30 30 30 10
(c)
3
8
9
16 25 5
+
=
+
=
= .
10 15 30 30 30 6
(d)
3
2
33 14 49
7
+
=
+
=
= .
7 11 77 77 77 11
(e)
3
x
33 7x 33 + 7x
+
=
+
=
.
7 11 77 77
77
(f)
3 2 3x 14 3x + 14
+ =
+
=
.
7 x 7x 7x
7x
3
Example 15. Recall that any integer, like 3, can be viewed as a fraction . Therefore
1
3 5
5 5 5
5 + 5 + 5 3 · 5 15
· = + + =
=
= .
1 7
7
7
7
7
7
7
� �� �
repeated addition
In other words, when we multiply fractions where one of them is really an integer, it’s the same as multiplying the tops
together.
In the following problems, reduce your answer. If the answer is “improper” (i.e. the top is bigger than the bottom)
leave it that way. Don’t rewrite it as a “mixed” fraction (i.e. something like 11/2).
(a) Multiply and reduce if possible: 4 ·
5
12
(b) Multiply and reduce if possible:
−5 7
·
1 10
(c) Multiply and reduce if possible:
1729
1
·
1
1729
Solution.
(a) 4 ·
5
20 5
=
=
12 12 3
(b)
−5 7
−35
7
·
=
=−
1 10
10
2
(c)
1729
1
1729
·
=
=1
1
1729 1729
CHAPTER 3. GROUP 1
34
Example 16. We saw in the last example how to justify multiplying the tops of fractions together. What about the
1
bottoms? We need to figure out what happens when we multiply by something like .
4
•
1
8
· 8 = = 2, which is one fourth of 8.
4
4
•
1
3
3
· 3 = . Is one fourth of 3? Yes:
4
4
4
3 3 3 3 4·3
+ + + =
=3
4
�4 4 �� 4 4�
add four times
3
So is one fourth of 3.
4
• For any number #, we have that
1
· # is one fourth of #.
4
So now we know:
1 1
1
· = one fourth of .
4 5
5
But we still haven’t figured out how to calculate it.
Start with a line, and break it into 5 pieces
one line
broken into 5 pieces
Now take one of those fifths, and break it into 4 pieces
five pieces
one fifth broken into 4 pieces
1 1
Each of these smallest pieces shows something of the size · . How big is this? Well, if we broke all of the fifths up
4 5
in the same way
then we see that we have 5 groups, and each group has 4 pieces. Thus, there are 5 × 4 = 20 total pieces. And one of
those pieces is one twentieth of the total. This justifies
1
1 1
· =
4 5 20
and from there we can multiply
1
times any fraction:
4
1 3
1 1
1
3
· = 3· · = 3·
= .
4 7
4 7
28 28
In the following problems, reduce your answer. If the answer is “improper” (i.e. the top is bigger than the bottom)
leave it that way. Don’t rewrite it as a “mixed” fraction (i.e. something like 11/2).
(a) Multiply the fractions and simplify if possible:
1 6
·
4 7
CHAPTER 3. GROUP 1
35
(b) Multiply the fractions and simplify if possible:
1 −10
·
5 11
(c) Multiply the fractions and simplify if possible:
1 8
·
10 9
Solution.
(a)
1 6
3
· = f rac628 =
4 7
14
(b)
1 −10 −10 −2
·
=
=
5 11
55
11
(c)
8
4
1 8
· =
=
10 9 90 45
Example 17. The previous two examples justify why we multiply the tops of fractions together, and why we multiply
the bottoms of fractions together. Now we just combine them:
3 7
1 3 7
1 3 · 7 1 21
21
21
·
= · ·
= ·
= ·
=
=
5 12 5 1 12 5 12
5 12 5 · 12 60
Of course we can simplify the last answer
21
3·7
7
=
= .
60 3 · 20 20
In the following problems, reduce your answer. If the answer is “improper” (i.e. the top is bigger than the bottom)
leave it that way. Don’t rewrite it as a “mixed” fraction (i.e. something like 11/2).
(a) Multiply the two fractions together, and reduce if possible:
1 10
· .
2 7
3 14
· .
4 9
x x
(c) Multiply the two fractions together, and reduce if possible: · .
2 7
(b) Multiply the two fractions together, and reduce if possible:
(d) Multiply the two fractions together, and reduce if possible:
Solution.
(a)
1 10 10 5
·
=
= .
2 7
14 7
(b)
3 14 42 7
·
=
= .
4 9
36 6
(c)
x x
x2
· = .
2 7 14
(d)
3x −13 3x(−13)
−39✁x
−39
·
=
=−
=−
.
2 5x
2(5x)
10✁x
10
3.4
3x −13
·
.
2 5x
Square roots
Example 18. Recall that (5 · 7)2 = 52 · 72 . Since this is true, a similar result holds for square roots:
√
(a) Simplify the following: 49
√
(b) Simplify the following: 121
√
5·7 =
√ √
5 · 7.
CHAPTER 3. GROUP 1
36
(c) Simplify the following:
√
(d) Simplify the following:
√
(e) Simplify the following:
√
(f) Simplify the following:
√
90
(g) Simplify the following:
√
49x (assume that x > 0)
(h) Simplify the following:
√
7x2 (assume that x > 0)
4·3
9·6
32
√
(i) Simplify the following: 75x36 (assume that x > 0)
√
Solution. (a) 49 = 7
√
(b) 121 = 121
√ √
√
√
(c) 4 · 3 = 4 3 = 2 3
√
√
√ √
(d) 9 · 6 = 9 6 = 3 6
√
√
√ √
√
(e) 32 = 16 · 2 = 16 2 = 4 2
√
√
√
√ √
(f) 90 = 9 · 10 = 9 10 = 3 10
√
√ √
√
(g) 49x = 49 x = 7 x
√
√ √
√
(h) 7x2 = x2 7 = x 7
√
√
√ √ √
√
(i) 75x36 = 25 · 3 · x36 = 25 x36 3 = 5x18 3
3.5
Factoring numbers
Example 19. Simplifying fractions like
√
30
and square roots like 90 both involve factoring numbers:
105
30
2 · 15 2
=
=
105 7 · 15 7
and
√
90 =
√
√
9 · 10 = 3 10
Recall that a prime number is one that we cannot factor any further.
(a) Factor 30 into primes.
(b) Factor 105 into primes.
(c) Factor 126 into primes.
(d) Factor 300 into primes.
(e) Factor 360 into primes.
(f) Factor 1470 into primes.
√
126
(g) Simplify √
1470
CHAPTER 3. GROUP 1
Solution.
37
(a) 30 = 3 · 10 = 3 · 2 · 5
(b) 105 = 5 · 21 = 5 · 3 · 7
(c) 126 = 2 · 63 = 2 · 3 · 21 = 2 · 3 · 3 · 7
(d) 300 = 3 · 100 = 3 · 2 · 50 = 3 · 2 · 2 · 25 = 3 · 2 · 2 · 5 · 5
(e) 360 = 2 · 180 = 2 · 2 · 90 = 2 · 2 · 2 · 45 = 2 · 2 · 2 · 5 · 9 = 2 · 2 · 2 · 5 · 3 · 3
(f) 1470 = 2 · 735 = 2 · 5 · 147 = 2 · 5 · 3 · 49 = 2 · 5 · 3 · 7 · 7
√
√
√
√
3 2·7
126
2 · 32 · 7
3 7
= √
=√
= √
(g) √
1470
7 2 · 3 · 5 7 15
2 · 3 · 5 · 72
3.6
Exponents
Example 20. Recall:
1
ab
√
b
means a
a−b means
a1/b
(an )m = anm
an am = an+m
an
= an−m
am
(ab)n = an bn
Using the above properties, simplify the following.
(a) (−2)5
(b)
x17
x22
(c) 4−3/2
√
(d) 36x4
Solution.
(b)
(a) (−2)5 = (−2)(−2)(−2)(−2)(−2) = −32
x17
1
= x17−22 = x−5 = 5
x22
x
1
1
1
= √
= 3=
3
2
8
( 4)
√
√ √
(d) 36x4 = 36 x4 = 6x2
(c) 4−3/2 =
1
43/2
Example 21. Simplify the following
�
(−2x−4 y6 )−8
(3x3 y−3 )−2
�−2
so that your final answer has no fractions, and each base, 2, 3, x and y, appears only once.
CHAPTER 3. GROUP 1
38
Solution. There’s more than one order you can do this in, and it really doesn’t matter too much which way you go.
But I think it does help to get some sort of a strategy and try to follow that. For instance, you could say “I’ll work from
the inside out, and simplify as I go.” Or you could say, “I’ll work from the outside in, and simplify at the end.” But
what you probably shouldn’t say is “I’ll randomly combine the inside and the outside, and move everything around
until I think of something to do with it.”
I’ll work from the inside out and simplify as I go:
�
(−2x−4 y6 )−8
(3x3 y−3 )−2
�−2
=
�
(−2)−8 x32 y−48
(3)−2 x−6 y6
�−2
�
�−2
= 2−8 3−(−2) x32−(−6) y−48−6
�
�−2
= 2−8 32 x38 y−54
= 216 3−4 x−76 y104