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Brooklyn College
Department of Mathematics
Precalculus Preparatory Workbook
Spring 2011
Sandra Kingan
Supported by the CUNY Office of Academic Affairs through funding for the Gap Project
ii
CONTENTS
1. Review of Pre-Algebra ............................................................................................................... 1
1.1. Fractions ............................................................................................................................... 1
1.2. Irrationals ............................................................................................................................. 3
1.3. Integer exponents ................................................................................................................. 7
1.4. Radicals .............................................................................................................................. 11
1.5. Properties of real numbers ................................................................................................. 12
2. Algebraic Thinking ................................................................................................................... 14
2.1. Algebraic expressions ........................................................................................................ 14
2.2. Linear equations ................................................................................................................. 16
2.3. Visualizing inequalities ...................................................................................................... 17
2.4. Solving inequalities ............................................................................................................ 19
2.5. Absolute value equations and inequalities ......................................................................... 20
3. Vizualization in Algebra ........................................................................................................... 22
3.1. Graphs of straight lines ...................................................................................................... 22
3.2. Finding the equation of a line ............................................................................................ 24
3.3. Functions and graphs ......................................................................................................... 25
1
1. REVIEW OF PRE-ALGEBRA
1.1. FRACTIONS
1) Simplify the fractions by writing them in lowest terms.
=6
3
=15
3
=10
4
=6
6
=5
4
=15
10
=30
27
=28
2
=36
8
2) Express each fraction as a mixed number.
=2
11
=5
16
=8
23
=3
25
=9
18
=7
20
=6
16
=4
34
=14
50
3) Express each mixed number as a fraction.
=5
21
=7
53
=5
210
=7
14
=9
55
=3
23
=3
220
=4
315
=3
131
2
4) Add/subtract fractions with like denominators. Leave your answer in lowest form.
=+4
1
4
2
=+3
2
3
1
=+7
5
7
6
=−5
1
5
4
=+2
1
2
1
=+9
2
9
3
=−9
3
9
6
=+9
8
9
7
=−6
2
6
4
5) Add/subtract fractions with unlike denominators. Leave your answer in lowest for
=+5
1
4
2
=+5
2
3
1
=+8
5
7
6
=−10
1
5
4
=+12
1
4
1
=+6
2
4
3
=−20
3
5
6
=+15
8
9
7
=−8
2
6
4
6) Multiply the fractions. Leave your answer in lowest form.
=×5
1
4
2
=×5
2
3
1
=×8
5
7
6
=)10
1)(
5
4(
=)12
1)(
4
1(
=)6
2)(
4
3(
=⋅20
3
5
6
=⋅15
8
9
7
=⋅8
2
6
4
7) Multiply the fractions. Leave your answer in lowest form.
=÷5
1
4
2
=÷5
2
3
1
=÷8
5
7
6
=
10
15
4
=÷12
1
4
1
=÷6
2
4
3
3
1.2. IRRATIONALS
1) Mark the specified decimals on the number line.
a) Mark every tenth point between 0 and 1.
b) Mark every tenth point between 0 and 0.1.
2) Compare the decimals. Write the symbols < , > , or = .
08.2_____5.2
09.0_____1.0
01.0_____1.0
05.1_____005.1
656565.1_____6565.1
656565.1_____6566.1
08.2_____5.2 −−
09.0_____1.0 −−
01.0_____1.0 −−
05.1_____005.1 −−
8888.1_____888.1
8888.1_____889.1
3) Write each repeating decimal using the bar notation.
Decimal Bar Notation
0.33333333… 3.0
0.16666666…
0.14287142857…
0.88888888…
0.09090909…
0.08333333…
0.285454545454…..
0 1
0 0.1
4
4) Write each fraction as a decimal using your calculator. List as many decimal places as
needed to recognize the pattern.
5.02
1=
3.03
1= 25.0
4
1=
2.05
1=
61.06
1= 142857.0
7
1=
=8
1
=9
1 =
10
1
=11
1
=12
1 =
13
1
=14
1
=15
1 =
16
1
=17
1
=18
1 =
19
1
5) Write each fraction as a decimal by guessing the pattern.
=2
1
=20
1 =
200
1 =
000,2
1
=3
1 =
30
1 =
300
1
=000,3
1
=4
1 =
40
1 =
400
1 =
000,4
1
=5
1
=50
1 =
500
1 =
000,5
1
=6
1
=60
1 =
600
1 =
000,6
1
5
=7
1
=70
1 =
700
1 =
000,7
1
=8
1
=80
1 =
800
1 =
000,8
1
=9
1
=90
1 =
900
1 =
000,9
1
=10
1
=100
1 =
000,1
1 =
000,10
1
6) Write each fraction as a decimal by guessing the pattern.
=10
2
=100
2
000,1
2
000,10
2
=10
3
=100
3 =
000,1
3 =
000,10
3
=10
4 =
100
4 =
000,1
4
=000,10
4
=10
5
=100
5 =
000,1
5 =
000,10
5
=10
6
=100
6
000,1
6
000,10
6
=10
7
=100
7
000,1
7
000,10
7
=10
8 =
100
8
000,1
8
000,10
8
=10
9
=100
9
000,1
9
000,10
9
6
7) What is the error when we approximate π by the fraction 7
22.
8) Draw a number line and mark the integers 0 to 10. Then mark π and e .
9) Fill in the table below using your calculator. Which square-roots are irrational?
Number Square root (at least up to 8 decimal places)
1
2
3
4
5
6
7
8
9
10
10) Write each decimal as a fraction in lowest form.
=1.0
=2.0
=3.0
=4.0
=5.0
=6.0
=7.0
=8.0
=9.0
=10.0
=11.0
=12.0
=25.0
=45.0
=50.0
=75.0
=95.0
=125.0
=375.0
=625.0
=875.0
7
1.3. INTEGER EXPONENTS
1) Fill in the table.
Exponential
Notation
Base Exponent Words
52 2 5 2 raised to the 5
3 4
-3 raised to 4
-2 4
-3 5
610
2) Write each multiplication expression as an
exponential notation.
a) 422222 =×××
b) =××××××× 33333333
c) =×× 888
d) =××××× 101010101010
e) =××× aaaa
f) =⋅⋅⋅⋅ 55555
g) =∗∗∗ 7777
h) =⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ yyyyyyyyyyyy
i) =⋅⋅⋅⋅⋅⋅⋅ xxxxxxxx
j) =−−−− )2)(2)(2)(2(
k) =−−−−− )6)(6)(6)(6)(6(
l) =−−− ))()(( xxx
3) Write each exponent as a multiplication
expression and simplify, if possible.
a) 16222224=×××=
b) 16)2)(2)(2)(2()2( 4=−−−−=−
c) =52
d) 5)2(−
e) =−4)3(
f) =53
g) =46
h) =510
i) =2
x
j) =−2)( x
k) =3y
8
4) Fill in the blanks.
_____20=
_____21=
_____22=
_____23=
_____24=
_____25=
_____26=
_____27=
_____28=
_____29=
_____210=
_____30=
_____31=
_____32=
_____33=
_____34=
_____35=
_____36=
_____37=
_____38=
_____39=
_____310=
_____40=
_____41=
_____42=
_____43=
_____44=
_____45=
_____46=
_____47=
_____48=
_____49=
_____410=
_____50=
_____51=
_____52=
_____53=
_____54=
_____55=
_____56=
_____57=
_____58=
_____59=
_____510=
5) Write each negative exponent as a positive exponent.
a) 4
4
2
12 =
−
b) 4
4
)2(
1)2(
−=−
−
c) =−52
d) 5)2( −−
e) =−−4)3(
f) =−53
g) =−46
h) =−510
i) =−2
x
j) =−3y
k) =−−5)( a
l) =−−4)5(
m) =−3)
2
1(
n) ( )
=−4
2
1
o) =−−2)
5
1(
p) =−2)
6
1(
9
6) Write the following numbers in scientific notation.
10 = 0.1 =
100 = 0.01 =
1,000 = 0.001 =
10,000 = 0.0001 =
100,000 = 0.00001 =
1,000,000 = 0.000001 =
10,000,000 = 0.0000001 =
100,000,000 = 0.00000001 =
1,000,000,000 = 0.000000001 =
10,000,000,000 = 0.0000000001 =
100,000,000,000 = 0.00000000001 =
7) Write each number in scientific notation.
_______________000,40 =
_______________000,000,000,9 =
____________ 0,000245,370,00 =
____________ 0.0004 =
____________ 005880.00000000 =
____________ 0.00053820 =
____________ 0.00700089 =
8) Write each number in standard form.
_______________107.5 6=×
_______________107.6 7=×
−
________________108.6 8=×
________________10034579.3 6=×
__________________102.3 5=×
−
__________________100056.0 7=×
9) Large and small number are everywhere in nature. Complete the following table.
Physics constant Number Scientific Notation
Speed of light in a vacuum 299,792,458 meters/second
Diameter of an atom 1110− meters
Diameter of the nucleus of an atom 1510− meters
Atomic mass 1510782,538,660.1 −×
Approximate age of the universe 13.7 billion years
Number of stars 22105×
10
10) Simplify using the exponential properties nmnmbbb
+= , nm
n
m
bb
b −= , ( ) nnn
baab = and n
nn
b
a
b
a=
a) =)2)(2( 54
b) =×31 33
c) =2653 3232
d) =32
xx
e) =7632 yxyx
f) =7632 yxyx
g) =7415 xyx
h) =5
4
2
2
i) =3
1
3
3
j) =3
2
x
x
k) =26
53
32
32
l) =36
72
yx
yx
m) =×5)32(
n) =
5
3
2
o) =5)(xy
p) =
5
y
x
q) =
5
3
2
a
xy
11) (Number sense) If the beginning of the line is 0 and the end is 1 billion, where would you
mark 1 million?
11
1.4. RADICALS
1) Find the values of the following radicals without using a calculator:
=4 =16 81 25.0
=3 8 =−3 8 =3 64 =−3 64
=4 16 =4 81 =4 128 =4 625
2) Find the values of the following radicals. Use your calculator.
=2 =54.7 =π 5.0
=3 10 =−3 10 =3 56.20 =3 027.0
=4 15 =4 45 =4 500 =4 16.0
6) Simplify using radical properties. Do not use your calculator.
a) =12
b) =18
c) =28
d) =48
e) =90
f) =300
12
1.5. PROPERTIES OF REAL NUMBERS
1) What are the following properties of real numbers. Think about the most elegant and concise
way of expressing these properties
a) The commutative property of addition
b) The commutative property of multiplication
c) The associative property of addition
d) The associate property of multiplication
e) The distributive property
f) The additive identity property
g) The multiplicative identity property
h) The additive inverse property
i) The multiplicative inverse property
2) Identify the property shown in each equation:
a) 2332 +=+
b) 2332 ×=×
c) 8)43()84(3 ++=++
d) 8)43()84(3 ××=××
e) )5(8)3(8)53(8 +=+
f) 909 =+
g) 65165 =×
h) 0)3(3 =−+
i) 15
15 =×
j) )42(3)4(3)2(3 +=+
13
3) Rewrite each expression using the distributive property. No need to simplify.
a) )8(7)6(7)86(7 +=+
b) =− )102(8
c) =− 6)25(
d) =+− )15(3
e) =−− 6)25(
f) =−−− )93(11
g) =−− )67(2
h) =−+− ))7(5(3
i) =+ )15(3
2
j) =+−
)15(3
1
k) =−−
)128(4
3
4) Answer the following questions.
a) What is the additive identity for real numbers?
b) What is the multiplicative identity for real numbers?
c) What is the additive inverse of -3?
d) What is the multiplicative inverse of 3
1?
e) What is the multiplicative inverse of 4
5−?
f) What is the additive inverse of 3
2?
g) What is the additive inverse of 5
1− ?
5) Give examples to show that the commutative and associative properties do not hold for
subtraction and division.
14
2. ALGEBRAIC THINKING
2.1. ALGEBRAIC EXPRESSIONS
1) Evaluate the algebraic expression 37 −x for the different values of x given below:
a) 3=x b) 15−=x c) 0=x
2) Evaluate the algebraic expression 85
−y
for the different values of y given below:
a) 3=y b) 15−=y c) 0=y
3) Express each phrase as an algebraic expression.
a) A number x times 5
b) 28 plus a number x
c) 28 times a number x
d) A number y minus 27
e) A number z increased by 19
f) A number z decreased by 25
g) Subtract a number x from 20
h) Subtract 20 from a number x
i) The product of 10 and a number z
j) A number divided by 17
k) 16 divided by a number x
l) Add 3 to a number x times 5
m) Add a number x to 20, then divide by 10
n) Subtract 5 from the product of x and 3
15
15
4) Expand each expression using the distributive property:
a) )8(7 +x
b) )10(8 −x
c) 6)5( x−
d) )1(3 +− x
e) )52(3 yx +
f) )1(7 ++ yx
g) )(3 wzyx +−+
h) )13(22 −+−− zyx
i) )1(3
1+
−x
j) )1(3
2+x
5) Simplify the algebraic expressions.
a) xx 32 +
b) xyx 375 +−
c) 5)1(2 ++− x
d) 1264 ++ xx
e) 3)2( ++− x
f) 5)42( +−− x
g) 32
xx+
h) )42(2
1−x
i) )4(3)5(2 −−+ xx
j) )7(82 yxyx +++
k) )7(823 yxyx −−+
l) )2(64 zxyxy ++
m) 4)1(5
3++x
n) )12()13(3 −−+ xx
0) )32(2
1)1(
5
3+++ xx
p) )32(2)2(9 +++− xx
16
16
2.2. LINEAR EQUATIONS
1) Solve the linear equations.
a) 72 =+x b) 13 =−x c) 35 −=−x d) 516 += x
2) Solve the linear equations.
a) 102 =x b) 9
4=
x c)
2
94 =− x d)
9
5
3
2−=
x
3) Solve the linear equations.
a) 743 =+x b) 5515 += x c) 439 −=− x d) 11 −=− x
5) Solve the linear equations.
a) 712
−=+x
b) 2
31 =+x c) 3
4
1
2
3=−
x d)
8
3
3
5
2
1=+− x
6) Solve the linear equations.
a) 10243 +=+ xx
b) 62510 +=+− xx
c) 64
59
4
3+=− x
x
d) 13
2
47 +=− x
x
e) 10
1
5
2
3
1
4−=+ x
x
f) 3
216
2
144 +=
− xx
g) 9
31
6
12 xx −=
+−
7) Solve the linear equations.
a) )5(243 +=+ xx
b) )11(4)3(2 +=+− xx
c) 43)1(24 +=++ xxx
d) 62)1(9 +=+− xx
17
17
2.3. VISUALIZING INEQUALITIES
1) Draw each inequality on a number line and write it as an interval (as shown):
a) 52 << x
h) 41 << x
b) 52 ≤≤ x
i) 13 ≤≤− x
c) 52 <≤ x
j) 64 <≤ x
d) 52 ≤< x
k) 03 ≤<− x
e) 2>x
l) 3>x
f) 2≤x
m) 2−≤x
g) ∞<<∞− x
n) 1−≥x
)5,2(
]5,2[
]5,2(
),2( ∞
]2,(−∞
),( ∞−∞
)5,2[
18
18
2) Draw each interval on a number line and write it as an inequality.
a) )1,3( −−
e) ]3,(−∞
b) )2,2(−
f) )3,( −−∞
c) ]3,1[−
g) ),2[ ∞
d) )3,3[−
h) ),1[ ∞−
3) Jane labels the interval below as )1,5( . Is she right? Explain.
4) Joshua labels the interval below as ]2,[−∞ . Is he right? Explain.
19
2.4. SOLVING INEQUALITIES
1) Solve the inequalities. Draw your answer on the number line and write it in interval form.
a) 05 ≤−x
b) 102 ≥x c) 2163 >+x
d) 0≤− x e) 3≤− x f) 102 ≤− x
g) 2042 −≤−− x h) xx −≤− 573 i) 34)3(
5
2+>+− xx
3) Solve the compound inequalities.
a) 412 <+≤ x b) 52
22 <+<
x
c) 7241 <+−<− x
20
2.5. ABSOLUTE VALUE EQUATIONS AND INEQUALITIES
1) Solve the following absolute value equations.
a) 2=x b) 65 =+x c) 8−=x
d) 512
3=+− x
e) 51 =−x f) 453 −=+ xx
g) 1752
1=−x
h) 21 +=− xx i) 542 −=+ xx
21
2) Solve the following absolute value inequalities.
a) 2≤x b) 5>x c) 51 ≤−x
d) 51 ≥−x
e) 625 <−x f) 1842 >+x
22
3. VIZUALIZATION IN ALGEBRA
3.1. GRAPHS OF STRAIGHT LINES
1) Draw a coordinate system. Plot the following points and state which quadrant they lie in.
)2,2(=A
)2,2(−=B
)2,2( −−=C
)2,2( −=D
)1,3(=E
)3,1(=F
)1,3(−=G
)5,2( −−=H
)2,4( −=I
)0,1(=J
)0,1(−=K
)1,0(=L
)1,0( −=M
)3,0(=N
)3,0( −=O
)0,5(−=P
2) Find the distance between the two points. Also find the midpoint.
a) )2,2( and )1,3(
b) )2,2( and )2,2(−
c) )2,4( − and )0,1(
d) )3,0( − and )0,5(−
3) Graph the following linear equations by plotting some points.
a) xy =
b) xy 2=
c) xy −=
d) 1+= xy
23
4) Graph the following linear equations by finding the x-intercept and y-intercept.
a) 1+= xy
b) 1+−= xy
c) 42 += xy
5) Graph the following horizontal and vertical linear equations.
a) 2=y b) 3−=y c) 4=x d) 1−=x
6) Find the slope (if it exists) of the line joining the two points.
a) )2,1( and )7,3(
b) )2,1( and )8,4(−
c) )2,1( and )5,1( −−
d) )2,1( and )4,1(
7) When the equation of the line is given in the slope-intercept form bmxy += we can read off
information as follows: slope is m and y-intercept is b. Practice reading this information from the
following equations.
a) 2+= xy
b) 12 +−= xy
c) xy 3=
d) 35
4−= xy
e) 12
3+−= xy
3.2. FINDING THE EQUATION OF A LINE
1) Find the equation of the line passing through the point P and with slope m.
a) )3,2(=P and 2=m
b) )3,2(=P and 2−=m
c) )2,1(=P and 7
5−=m
d) )1,3( −=P and 0=m
e) )1,3( −=P and slope is undefined
3) Find the equation of the line passing through the points P and Q.
a) )1,3( −=P and )5,2(=Q
b) )5,0(=P and )5,3(=Q
4) Determine if the lines are parallel, perpendicular, or neither
a) 12 += xy and 52 += xy
b) 12 += xy and 52 +−= xy
c) 12 += xy and 52
1+= xy
d) 12 += xy and 52
1+−= xy
e) 13
2+= xy and 5
2
3+−= xy
f) 0432 =++ yx and 0823 =++− yx
g) 2=y and 0=x
h) 2=y and 3−=x
5) Find the equation of the line that is parallel to and the line that is perpendicular to the given
line and passing through the given point.
a) 12 += xy and )2,1(=P
b) 13
2+−= xy and )4,3( −−=P
c) 3=y and )5,1(=P
25
3.3. FUNCTIONS AND GRAPHS
1) Determine whether the correspondence is a function.
a) b) c) d) e) f)
2) Determine whether the relation is a function. Identify the domain and range.
a) )}20,4(),15,3(),10,2{(
b) )}1,7(),1,5(),1,3{(
c) )}7,0(),4,2(),1,2(),3,7{( −−−
d) )}9,1(),7,1(),5,1(),3,1{(
e) )}1,3(),1,4(),1,2(),1,0(),1,2{( −−
f) )}2,3(),1,5(),0,0(),1,3(),0,5{( −−−
3) Find the specified function values:
a) xxf 3)( = =)1(f _____ =− )4(f _____ =)0(f _____ =)(af ______
b) 26)( −= xxg =)1(g ____ =− )4(g _____ =)0(g _____ =)(ag ______
c) 4)( 2−= xxh =)1(h ____ =− )4(h _____ =)0(h ______ =)(ah ______
4) Find the specified images and preimages:
a) xxf 3)( = image of 5 is _____ pre-image of 20 is _______
b) 26)( −= xxf image of 5 is _____ pre-image of 20 is _______
c) 4)( 2−= xxf image of 5 is _____ pre-image of 20 is _______
26
5) Use the vertical line test to determine if y is a function of x. If it is a function use the
horizontal line test to determine if it is a one-to-one function.
2xy = 13
−= xy
2yx = 922
=+ yx
12−= xyx yx =
27
6) Find the domain and range of the following functions. Where is the function increasing,
decreasing or constant? (Note that the ends go off to infinity even though there are no arrows.
This is because graphing software does not put arrows.)
2)( xxf = 10)( 2
+= xxf
153)( 23+++= xxxxf 24 63)( xxxf −=
1)( −= xxf 1)( −= xxf
x
xxf =)( 225)( xxf −=
