G P S F O R MAT 0 6 5 0 A Workbook of Problems for College ...
Transcript of G P S F O R MAT 0 6 5 0 A Workbook of Problems for College ...
TABLE OF CONTENTS
ββBegin at the beginningβ, the king said, very gravely, βand go till you come to the end:
then stopββ.βLewis Carroll
Preface
I Chapter 1: Introduction
Section 1.1: Nomenclature-Terms, Constants, Monomials, Binomials,
Trinomials, Absolute Values
Section 1.2 Exponents- aN Γ aM; (ab)N; (a/b)N; aN/aM; a0 ; a-N
Section 1.3 Scientific Notation
II Chapter 2: Signed Numbers
Section 2.1 Operations involving Multiplication; Division; Addition;
Subtraction
Section 2.2 Order of Operations
III Chapter 3: Evaluation
Section 3.1 Evaluating Algebraic Expression
Section 3.2 Functional Notation
IV Chapter 4: Multiplication of Algebraic Expressions
βI wish I didnβt know now, what I didnβt know then.β
Section 4.1 Multiply Monomial X Monomial X Monomial
Section 4.2 Monomial X Binomialβthe Distributive Law
Section 4.3 Monomial X Trinomial
Section 4.4 Binomial X Binomial; (Binomial)2
Section 4.5 Binomial X Trinomial; (Binomial)3
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V Chapter 5: Combining Like Terms
VI Chapter 6: Factoring
Section 6.1 Unique Prime Factorization of Integers
Section 6.2 Greatest Common Monomial Factor (GCF)βNumerical; Variable
Section 6.3 Difference of Two Squares (DOTS)
Section 6.4 Combination of Greatest Common Monomial Factor and Difference
of Two Squares
Section 6.5 Trinomial with Leading Coefficient 1 (a=1): ax2 + bx + c
Section 6.6 Trinomial with Leading Coefficient Other than 1 (aβ 1):
ax2 + bx +c
Section 6.7 Combination of Greatest Common Monomial Factor and
Trinomial
Section 6.8 Factoring by Grouping
Section 6.9 Review of All Factoring
VII Chapter 7: Solving Equations and Inequalities
Section 7.1 Introduction to Equations- Is a Given Number a Solution
Section 7.2 Linear (First Degree) Equations
Section 7.3 Decimal Equations
Section 7.4 Fractional EquationsβMonomial in the Numerator; Binomial in
the Numerator
Section 7.5 Literal Equations
Section 7.6 Quadratic (Second Degree) Equations
2
Section 7.7 InequalitiesβShowing Solutions algebraically and on the number
line
VIII Chapter 8: Fractions
Section 8.1 Simplifying FractionsβMonomial/Monomial;
Polynomial/Monomial; Polynomial/Polynomial
Section 8.2 Multiplying Fractions
Section 8.3 Dividing Fractions
Section 8.4 Combining FractionsβLike Denominators; Different
Denominators
IX Chapter 9: Roots and Radicals
Section 9.1 Pythagorean Theorem
Section 9.2 Simplifying RadicalsβNumerical; Variable
Section 9.3 Multiplying RadicalsβMonomial Γ Monomial; Monomial Γ Binomial
Section 9.4 Dividing Radicals- βπΉππππ‘πππ; Radical Divided by a Radical
Section 9.5 Combining Radicals- Like Radicands; Unlike Radicands
X Chapter 10: Graphing Linear Functions
Section 10.1 Introduction- The Rectangular Coordinate System; Plotting
Points
Section 10.2 Table Method
Section 10.3 Intercepts Method
Section 10.4 Slope β Y Intercept Method
XI Chapter 11: Solving a System of Two Equations In Two Unknowns
Section 11.1 Solving Graphically
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Section 11.2 Solving by the Substitution Method
Section 11.3 2 Solving by the Elimination Method
XII Chapter 12: Verbal Problems
Section 12.1 Ratio and Proportion Problems
Section 12.2 Percent Problems
Section 12.3 Number Problems
Section 12.4 Distance = Rate Γ Time Problems
Answers to the Odd Numbered Problems
βWhere of one cannot speak, there of one must be silent.β
PREFACE
Many years ago, a man in Australia made a bet with a friend that he could eat
an entire Dodge Dart automobile in less than one yearβs time βthis was excluding the
rubber and glass parts, of course. When asked how he thought he could do it
without getting sick and dying, the man replied that he would cut the car into very,
very small pieces and eat it that way. The man was able to accomplish this task and
won his wager.
In this textbook, we will follow the βDodge Dartβ approach. The mathematical material will be presented in very, very small pieces. It is hoped that in this way, you
will be able to easily digest the mathematical material presented.
It is our objective that you obtain a great understanding of the elementary
algebra presented herein. If you donβt obtain a great understanding, it is hoped that
you receive a small understanding of the material. If you donβt receive a small understanding, it is hoped that at least it doesnβt make you sick.
Enjoy
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Section 1.1: Nomenclature--Terms, Constants, Monomials,
Binomials, Trinomials, and Absolute Values
βAlgebra is the use of symbols to represent math things: numbers, lines, unknowns; and it is the study of rules for combining/working with these
symbols.β
Identify the coefficients and variables in each of the following: 1) 8π₯2 2) β2π¦3
3) 7π₯2π¦ 4) π¦π§ 5) 12π₯π¦π§ 6) 2π₯2 + 4
Classify each of the following as monomials, binomials, or trinomials: 7) 2π₯ 8) 6π₯2 β 9 9) 3π₯2 10) π₯8 + 45 11) 4π₯2 + 7π₯ β 11 12) π₯4 β 3π₯2 + π₯
Identify each term in the following algebraic expressions: 13) 4π₯2 + 8π₯ β 10 14) 2π₯4 β 9 15) β6π₯3 + 7π₯2 β 3π₯ + 4 16) 19π₯4 β 12π₯3 + 10π₯2 β 5π₯ + 1
Identify the variable terms and the constant terms in the following 17) 10π₯2 β 7π₯ + 2 18) 6π₯3 + 10 19) β7π₯4 + 3π₯2 β 4π₯ + 17 20) 9 β 10π₯ + 4π₯2
Identify the degree, number of terms, and constant terms for: 21) 3π₯2 + 8π₯ β 7 22) 4π₯5 + 2π₯ + 3 23) 9π₯16 β 4π₯12 β 5π₯ β 21 24) 4π₯8
24) 2π₯7 β 4π₯6 + 9π₯5 + 6π₯3 β 7π₯2 26) 2π₯ β 24
Evaluate each of the following 27) |β4| 28) |9| 29) |β24| 30) |41|
31) |3| 32) |
β4|
8 5 33) β|11| 34) β|β12|
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Section 1.2: Exponents
βThe only way to learn mathematics is to do mathematics. The problem is not
the problem. The problem is your attitude about the problem.β
Simplify:
1) 32 Γ 34
3) (β4)2
5) β42
7) π₯2 β π₯6
9) π5 Γ π6
11) (π7π9)(π3π6)
13) (5π₯2π¦3)(β4π₯5π¦2)
15) β7ππ2(β3π2π3)
17) 2(π₯2)3
19) (3π§4)4
21) (2π2π3)4
π6
23) π4
π5
25) π
π3
27) π9
2) 23 Γ 24
4) (β4)3
6) β43
8) π¦4 Γ π¦8
10) (π₯3π¦4)(π₯2π¦6)
12) (π2π 3π‘) Γ (π3π 4π‘2)
14) β6ππ2(3π3π)
16) (2π₯2)3
18) (5π¦3)2
20) (π5)6
22) (5π4π6)3
π7
24) π2
π4
26) π7
π7
28) π12
6
29) πβ2
π‘β3
31) π‘5
(π€5)3
33) π€2
π¦4
35) (π¦3)5
37) (3β4)β2
39) (5β6)β2
41) (β4)β2
43) β(β4)β2
45) πβ3 Γ π4
47) (4πβ3)2(2πβ4)3
49) (β5π₯2)4
51) (5π₯2π¦β4)3
53) 2π2(4π2)(β2π)
π β3
30) π 4
32) π£5
π£5
(π₯2)6
34) π₯5
π§β5
36) (π§β3)4
38) (2β3)2
40) 4β2
42) β4β2
44) πβ4 Γ πβ6
46) (π3πβ5)3
48) (2π2πβ4)β3
50) β5(π₯2)4
52) 6π₯3(4π₯2)(β2)
54) 7π(3π2)(β1)
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Section 1.3: Scientific Notation
βLook not at the flask, but at what it contains.β
Convert into Scientific Notation 1) 2,400 3) 280 5) .003 7) .00000045
Convert into standard form 9) 4.8 Γ 105
11) 8.14 Γ 109
13) 9.4 Γ 10β2
15) 3.56 Γ 10β1
2) 3,600,000 4) 410,000 6) .05 8) .000874
10) 7.43 Γ 103
12) 3.14 Γ 102
14) 1.44 Γ 10β5
16) 6.18 Γ 10β8
Multiply and leave answer in standard form 17) (2 Γ 104)(3 Γ 103) 18) (6 Γ 107)(4 Γ 105) 19) (8.1 Γ 1012)(5.2 Γ 104) 20) (4.3 Γ 103)(3.6 Γ 108) 21) (4 Γ 10β2)(2 Γ 10β6) 22) (7 Γ 104)(3 Γ 10β10) 23) (6.9 Γ 10β2)(3.4 Γ 10β6) 24) (4 Γ 103)(4 Γ 10β3)
Divide and leave answer in standard form (6Γ104)
25) 26) (2Γ102)
(9Γ10β6) 27) 28)
(6Γ10β3)
(4.6Γ103) 29) 30)
(2.3Γ10β4)
(8.4Γ106)(5.1Γ104) 31) 32)
(4.2Γ105)
(1.4Γ109) 33) 34)
(2.8Γ104)(5Γ1012)
(8Γ105)
(5Γ10β6)
(4Γ10β7)
(5Γ102)
(3.4Γ104)
(1.7Γ109)
6.4Γ1015)(9Γ104)
(4.5Γ108)
(4.8Γ10β8)
(6.1Γ10β2)(2.4Γ10β11)
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Section 2.1: Operations Involving Addition, Subtraction,
Multiplication or Division
βThere are three types of students: those who can count and those who canβt.β
Multiply:
1) 5(6) 2) 4 Γ 3 3) β7(3) 4) β6 Γ 4 5) (β2)(5) 6) 3(β10) 7) 5 Γ (β9) 8) (7)(β8) 9) (β7)(β3) 10) β8(β4) 11) β11 Γ (β3) 12) 4(2)(5) 13) 3(β2)(6) 14) 7(β3)(β2) 15) β8(β4)(β5) 16) β3(β2)(β8)(β1)
Divide:
36 17)
6
48 18)
12
β24 19)
3
β16 20)
8
22 21)
β11
50 22)
β5
β12 23)
β6
β18 24)
β9
25) 63 Γ· 7 26) β30 Γ· 15
27) 28 Γ· (β14) 28) (β100) Γ· (β10)
0 29)
9
7 30)
0
9
Combine (Add or Subtract):
31) 4 + 6 32) 8 + 7 33) β9 + 14 34) β11 + 6 35) 12 + (β5) 36) 19 + (β8) 37) (β10) + (β11) 38) (β17) + (β7) 39) 17 β 5 40) 14 β 6 41) 12 β (β8) 42) 9 β (β6) 43) β3 β 4 44) β2 β 11 45) β10 β (β5) 46) β8 β (β7) 47) 4 + 8 + 6 48) 14 + 9 + (β5) 49) 10 + (β6) + (β9) 50) β12 + (β7) + (β3) 51) β15 + 8 + 7 52) β19 + (β10) + 8 53) β9 + 12 + (β3) 54) 2 + 6 β 4 55) 7 + 10 β (β5) 56) 18 β 7 + 6 57) 22 β (β3) β (β7) 58) β13 β 9 β (β11) 59) 4 + (β8) β (β7) + (β6) 60) 10 β (β2) β (4) + (β10) 61) β12 + (10) β (β9) β (3) 62) β15 β (β7) β (β9) β (β5) + 9
10
Section 2.2: Order of Operations
βMathematics is not a spectator sport. You have to do it to appreciate it, and doing it
Evaluate:
1) 4 + 5(6)
3) 8 + 4(β2)
5) 7 β 4 Γ· 2
7) (2 + 3)4
9) 5 β 23
11) 8 β 16 Γ· 23
13) 35 Γ· 5 + 4 Γ 3
15) 22 β 5 Γ 2 + 24
17) 10 β 6(5 β 2)β9 (42β9)
19) (β7+4(0)β69β3)
(β4β8) 21)
(β5β1)
(β3(β5)β(β6)(β4) 23)
β3(8β5)
(β5β(β3)2) 25)
β7
requires patience and persistence.β
2) β2 + 3(5)
4) β8 β 5(β2)
6) 15 β 10 Γ· 5
8) (10 β 8)6
10) β4 β 32
12) 18 β 32 Γ· 42
14) 20 Γ· 2 β 32 β 4 β 2
16) 24 + 14 Γ· 7 β 8
18) 3(β4)2 + 25β(8 β 3)2
(25+6) 20)
(β2+4(2)β6)
(β14+(β6)) 22)
2(β5)
(4(β7)+2(7β3)) 24)
4(β5)
26) (12+3)
β (2β6)2
(4+1) 4
11
(32β(7β8)3) 42β6 27) 28)
5(β2)+4(6β3) β5β3(8β2)
(β2)3β3(β4) β2(5β6)2
29) 30) 2(β2) β6+4β3
4β32 10β22
31) β8(5 β 3) + 32) 10(9 β 4) + 5 2(β3)
33) 5(2 β 6)2 + 7 Γ 0 β (4 β 5)2 34) 11 β 4(3 + 2) β 6 β 0 β (β2)3
35) 20 β 24 Γ· 8 + 6 36) 6 β 15 Γ· 3 Γ 5
37) 35 Γ· 7 Γ 3 β 5 38) 40 Γ· 23 Γ 3 β 9
39) 8(β7 β 5) 40) 8 β 7(β5)
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Section 3.1: Evaluating Algebraic Expressions
βAlgebra symbols are used when you donβt know what you are talking about.β
Evaluate:
1) 4π₯ π€βππ π₯ = 3
2) β2π₯ π€βππ π₯ = 5
3) β6π₯ π€βππ π₯ = β3
4) 7π₯2 π€βππ π₯ = 2
5) 3π₯2 π€βππ π₯ = β4
6) 2π₯2π¦ π€βππ π₯ = 4, π¦ = β2
7) 4π₯2π¦2 π€βππ π₯ = β3, π¦ = β1
8) 2π₯3 π€βππ π₯ = β2
9) π₯4 π€βππ π₯ = β1
10) 4π₯π¦π§ π€βππ π₯ = 2, π¦ = 3, π§ = β4
11) β2π₯2π¦π§ π€βππ π₯ = 1, π¦ = β2, π§ = 4
12) 6π₯2π¦π§2 π€βππ π₯ = 2, π¦ = 4, π§ = 1
13) 3π₯ + 4π¦ π€βππ π₯ = 5, π¦ = β6
14) 5π₯ β 2π¦ π€βππ π₯ = 4, π¦ = β3
15) 2π₯ β 3π¦ + 7π§ π€βππ π₯ = 3, π¦ = 2, π§ = β4
(π₯2β2π¦2) 16) π€βππ π₯ = 4, π¦ = 3
2π₯
13
(π₯3β3π¦2+2) 17) π€βππ π₯ = 4, π¦ = 2
3π¦
(4βπ₯2) 18) π€βππ π₯ = 8, π¦ = 5
4π¦
19) 2π₯2 β 3π₯π¦ + π¦2 π€βππ π₯ = 4, π¦ = 2
20) 4π₯2 β 2π₯π¦ β π¦2 π€βππ π₯ = β2, π¦ = 3
21) 2π5π2 π€βππ π = β1, π = β2
(4βπ₯2) 22) π€βππ π = 5, π = 2
4π¦
(2ππβπ2+1) 23) π€βππ π = 4, π = 1
2π
(4ππ+π+10) 24) π€βππ π = 5, π = 2
βπ
25) 2π3π2 β 4π3 π€βππ π = β1, π = β2
26) 3π₯2 + 4π¦3 + 5π§4 π€βππ π₯ = 3, π¦ = 2, π§ = 1
27) 2π₯ β 3π¦ + 4π§3 π€βππ π₯ = 5, π¦ = β2, π§ = β1
28)5π β 4π + 3π β 2π π€βππ π = β8, π = β6, π = β4, π = β2
14
Section 3.2: Functional Notation
βThereβs no such thing as talent; you just have to work hard enough.β
Evaluate:
1) π(5); π(π₯) = 4π₯ β 3
2) π(4); π(π₯) = 6 β 8π₯
3) π(β2); π(π₯) = 5 β π₯2
4) π(3); π(π₯) = π₯2 β 2π₯ + 6
5) π(β1); π(π₯) = π₯2 + 4π₯ β 9
6) β(β2); β(π₯) = 3π₯2 β 4π₯ β 5
7) πΉ(β3); πΉ(π₯) = β2π₯2 + 5π₯ β 6
8) πΊ(4); πΊ(π₯) = βπ₯2 β 7π₯ + 2
9) π»(3); π»(π₯) = β4π₯2 + 3π₯ β 7
10) β(2); β(π₯) = π₯3 + π₯2 β 2π₯ + 6
11) π(1); π(π₯) = β5π₯3 + 8π₯2 β 3π₯ + 4
12) π(β1); π(π₯) = 2π₯4 + 4π₯3 β 6π₯2 + 9π₯ β 10
13) πΉ (1) ; πΉ(π₯) = 10π₯ + 4
2
14) πΉ (1) ; πΉ(π₯) = 6π₯ β 1
3
(π₯+3) 15) β(3); β(π₯) =
(π₯β2)
15
(π₯2+1) 16) π(3); π(π₯) =
π₯
(π₯2+2π₯+3) 17) π(1); π(π₯) =
2π₯
18) πΉ(β2); πΉ(π₯) = (π₯ + 4)(π₯ + 7)
19) πΉ(1); πΉ(π₯) = (π₯ β 1)(π₯ + 2)(π₯ β 3)(π₯ + 4)
20) π(2); π(π₯) = π₯4 β 1
21) πΊ(1); πΊ(π₯) = π₯3 + π₯2 + 4π₯ + 8
22) πΊ(β1); πΊ(π₯) = π₯3 β 2π₯2 + 3π₯ β 7
23) π(0); π(π₯) = π₯9 + 3π₯7 β 5π₯2 + 2
24) π(π); π(π₯) = 6π₯ + 2
25) π(π + β); π(π₯) = 4π₯ + 3
26) πΉ(π); πΉ(π₯) = π₯2 + 8π₯
27) πΉ(π + β); πΉ(π₯) = π₯2 + 4π₯ + 6
28) πΊ(π); πΊ(π₯) = 2π₯2 β 3π₯ β 6
29) π(π); π(π₯) = β4π₯2 + 2π₯ β 6
30) π(π + β); π(π₯) = β3π₯2 β 6π₯ + 8
16
Section 4.1: Multiply MonomialβMonomial
βI will guard everything within the limits of my post and quit my post only when properly relieved.β
Multiply: 1) π₯2 β π₯3
2) π¦3 β π¦4
3) 4π2 β 5π6
4) (β4π4)(8π3) 5) (3π 5)(β6π 7) 6) (β2π‘2)(β5π‘3) 7) (3ππ2)(8π2π) 8) (4π2π3)(β6ππ2) 9) (β2π4π )(5π2π 3) 10) (β6π₯5π¦3)(β8π₯2π¦4) 11) (3π₯4π¦5)2
12) (β4π4π6)2
13) 4π₯2(4π₯)2
14) (π₯π¦2)(π₯π¦)2
15) (9π₯π¦2)(2π₯2π¦)(3π₯π¦) 16) (4ππ3)(2π2π4)(β3π5π2) 17) (6ππ 5)(β3π4π 3)(β5π2π 4) 18) (β5π₯π¦2)(β2π₯3π¦4)(β8π₯6π¦5) 19) (15π₯2π¦3π§4)(4π₯π¦2π§5) 20) (β7π₯2π¦3π§)(9π₯3π¦8π§10) 21) (β8π2π6)(β3π3π8) 22) (4π2π 3π‘5)(4ππ 2π‘4)(3π3π π‘6) 23) (β2π₯2π¦3π§)(3π₯π¦4π§2)(4π₯2π¦2π§5) 24) (8π2π2π5)(β4π3π4π7)(β3ππ5π6) 25) (β2π5π4π3)(β3π4π3π6)(β5π2π5π6) 26) 3π₯π¦(4π₯π¦)2
27) β2ππ2(3π2π3)2
28) β4π₯(2π₯2π¦3)2(5π₯2π¦) 29) (2π₯π¦)(3π₯2π¦)2(10π₯2π¦) 30) (3π₯2π¦)2(2π₯3π¦4)2
31) (2π₯3)(4π₯2)(β5π₯)
17
Section 4.2: Multiply MonomialβBinomial-The Distributive Law
βIn theory, thereβs no difference between theory and practice, but in practice there is.β
Multiply:
1) 8(2π₯ + 6) 2) β5(4π₯ + 10) 3) 6π₯(3π₯2 + 7) 4) π₯2(π₯ + 4) 5) β2π¦2(π¦2 + 3) 6) 4π¦3(2π¦4 β 6π¦) 7) β9π§2(3π§2 β 5π§) 8) β4π§3(8π§2 + 2π§) 9) (2π + 5)π3
10) β5π2(3π β 1) 11) 4π(11π β 2π2) 12) 6π(π2 + 4π β 8) 13) β2π(π2 β 10π + 9) 14) β6π 2(4π 2 + 2π β 3) 15) π₯2π¦(3π₯2π¦-4xπ¦2 + 9) 16) 2π2π3(4π2π β 5ππ β π) 17) (π₯2 + 5π₯ β 6)(β2π₯2) 18) 2π₯(π₯3 + 4π₯2 β 10π₯ + 7) 19) β4π4π 3(5π3π 2 + 6π2π + 4π2 β 1) 20) 3π₯3(5π₯4 + 3π₯3 β 2π₯2 + 9π₯ β 10)
18
Section 4.3: BinomialβBinomial
βBe kind, for everyone you know is fighting a hard battle.β
Multiply: 1) (π₯ + 1)(π₯ + 3) 2) (π₯ β 2)(π₯ + 4) 3) (π₯ + 7)(π₯ β 5) 4) (π₯ β 6)(π₯ β 8) 5) (8 + π¦)(2 + π¦) 6) (4 + π¦)(9 β π¦) 7) (3 β π¦)(2 + π¦) 8) (10 β π¦)(4 β π¦) 9) (2π + 3)(π + 9) 10) (5π + 6)(2π β 3) 11) (4π β 3)(3π + 5) 12) (8 + 5π)(3 β 2π) 13) (6π β 5)(2π β 3) 14) 2(π₯ + 3)(π₯ β 9) 15) 5(3π₯ β 2)(π₯ β 1) 16) 2π₯(7π₯ + 1)(2π₯ + 3) 17) 3π₯2(6π₯ + 1)(π₯ β 4) 18) (π₯ + 1)2
19) (π₯ β 3)2
20) (2π₯ + 5)2
21) (3π₯ β 7)2
22) (5 β 3π₯)2
23) (π₯ + 4)(π₯ β 4) 24) (π₯ + 9)(π₯ β 9) 25) (6 + π¦)(6 β π¦) 26) (5 β π§)(5 + π§) 27) (4π₯ + 5)(4π₯ β 5) 28) 3π₯(π₯ + 2)2
29) 5π₯(2π₯ + 1)2
30) 6π₯2(π₯ + 4)2
19
Section4.4: Multiply BinomialβTrinomial or TrinomialβTrinomial
βSo much that happens, happens in small ways.β
Multiply:
1) (π₯ + 1)(π₯2 + 2π₯ + 3) 17) (π₯2 + 1)(π₯2 + 5π₯ + 7)
2) (π₯ + 2)(π₯2 β 5π₯ + 6) 18) (π₯2 + 2π₯ + 1)(π₯2 β 4π₯ + 4)
3) (π₯ β 3)(π₯2 β 2π₯ + 8) 19) 2π₯(π₯ + 5)(π₯2 + 4π₯ β 12)
4) (π¦ + 8)(π¦2 + 4π¦ + 4) 20) (π₯ + 1)2(π₯ β 2)2
5) (π¦ β 4)(2π¦2 + 5π¦ β 12)
6) (3π¦ β 5)(π¦2 + 7π¦ + 10)
7) (5π§ β 6)(π§2 + π§ + 8)
8) (7π§ + 3)(3π§2 β 13π§ β 5)
9) (π§ + 9)(4π§3 + 6π§ β 3)
10) (π + 10)(5π3 β 4π2 + 2π)
11) (π + 7)(π + 1)2
12) (π β 3)(π + 2)2
13) (π + 1)3
14) (π + 2)3
15) (2π + 1)3
16) (3π β 2)3
20
Section 5.1: Combining Like Terms
Combine Like Terms: 1) 4π₯ β 6π₯ β 7π₯ 2) 5π₯π¦ + 8π₯π¦ β 4π₯π¦ 3) 7π₯2π¦ β 2π₯2π¦ + 3π₯2π¦ 4) 6(π₯ + 4) β 5π₯ 5) 9(π₯ + π¦) β 7(π₯ β π¦) 6) 8(2π + 3π) β 4(5π β 2π) 7) 3(8π β 2) β 2(4π β 1) + 7(π β 4) 8) 10ππ + 5π2π β 4ππ 9) 11ππ 2 + 4 β 2ππ 2 + 6 10) 4ππ 2 + 6π2π β 2ππ 11) 2π₯π¦(3π₯ + 5π¦) β 4π₯(π₯π¦ + 2π¦) 12) 6π2π β 2π(ππ 2 + 5ππ ) β 6ππ 2
13) 4π3 β 5π(3π2 + 2π) β 7π(2π + 1) 14) 2(π₯ + 4) β π₯(π₯ β 3) + 10π₯2
15) (2π₯2 + 3π₯ β 4) + (5π₯2 β 8π₯ β 2) β (3π₯2 β 6π₯ β 1) 16) (8π¦2 β 5π¦ + 3) + (2π¦2 + 6π¦ β 9) β (π¦2 β 7π¦ β 11) 17) 3π₯2(π₯ β 4) β π₯(3π₯ β 5) β 16 18) β5π π‘(2π 2π‘2 β 3π π‘) + 10π 3π‘3
19) 5π₯(π₯ + 4) β 2(π₯2 β π₯) β 3(π₯ + 9) 20) 3π¦(π¦ + 1) β 6(π¦2 + π¦) β (π¦ β 7) β 4 21) β3ππ(2π2 β 3ππ + 4ππ2) + 5π2π 22) 5π β (3π β π) β π 23) 9 β 7(2π₯ β π¦) + 7π¦ 24) (2π₯2π¦ + 5π¦) β (3π₯2π¦ β 2π¦) 25) (8π₯2 β 3π₯ β 9) β (4π₯2 + 2π₯ β 7) 26) 4π₯2(2π₯ β 3) β 5π₯(π₯2 + 6π₯) 27) β4ππ (6π2π 2 β 5ππ ) β 10π2π 2(ππ + 2) + 10ππ 28) 2ππ2 β 3π(ππ2 + 5π2π) + 9π2π2
29) 6π2π β 2π(ππ2 β 5ππ) β 7π2π2
30) 8 β 2(5π₯2 β 4) β 3(2π₯2 β 1) β (π₯2 β 5) 31) 5π₯(2π₯2π¦2 β 4) β 2(π₯3π¦ + 3π₯) + 2π₯2
32) 2π₯6(4π₯3 β 3π¦) β 4π₯(π₯8 β 2π₯5π¦ + 5(π₯9 + π¦) 33) (8π¦2 β 4π¦ β 3) β (2π¦2 β 3π¦ β 7) β (6π¦2 + π¦ β 4) 34) 3(4π§2 β 2π§ β 1) β 5(3π§2 β 4π§ β 2) + 2(π§2 β 2π§ β 9) 35) 6 β 5(π₯2 + 4π₯ β 3) β 3(2π₯2 β 5π₯ β 2) β 9(3π₯2 β 6π₯ + 8)
21
Section 6.1: Unique Prime Factorization of Integers
βSo we beat on, boats against the current, borne back ceaselessly into the past.β
Factor into prime factors:
1) 12 29) 360
2) 24 30) 1440
3) 96
4) 144
5) 18
6) 98
7) 20
8) 200
9) 72
10) 36 11) 99
12) 120
13) 48
14) 54
15) 80
16) 500
17) 108
18) 40
19) 42
20) 75
21) 27
22) 88
23) 480
24) 105
25) 250
26) 245
27) 720
28) 880
22
Section 6.2: Greatest Common Monomial Factor (GCF)-Numerical or Algebraic
βAs my pappy always used to say: there ainβt no horse that canβt be broken and there ainβt no rider that canβt be throwed.β
Factor Completely:
1) 2x β 4 2) 8y + 16 3) 15π§2 + 8π§ 4) 3π₯2 + 6π₯ + 12 5) 4π¦3 + 20π¦2 β 8π¦ 6) π4 + π3 β π2 + 5π 7) 7π3 β 14π2 + 21π 8) 4π2π2 β 2ππ + 6π2π β 8ππ2
9) 36x2y3 + 12xπ¦3
10) 36π3π4 + 18π2π3 + 12ππ2 + 6ππ 11) 14r3 β 28π2 + 35π 12) π 6 + 2π 4 β π 2
13) 20π10 + 10π5 β 5π3
14) 2π‘3 β 4π‘2 + 8π‘ 15) 18π₯3π¦4π§5 β 27π₯2π¦3π§3 + 9π₯2π¦2π§2
16) 12π¦7 β 9π¦5 + 3π¦2
17) 11π2ππ β 22π2π2π + 33π2π2π2
18) 2π₯2 β 3π¦2 β 4π§2
19) 2π₯π¦π§2 β 3π₯π¦π§ + 4π¦π§ 20) 4πππ + 5ππ β 6ππ 21) 4π₯3 + 8 22) 12π4 + 48π3 β 96π2
23) 15π₯π¦2 + 30π₯π¦3
24) 10π2π + 20ππ2 β 30ππ 25) 6π₯2 + 16π¦2 + 26π§2
26) 36π4 + 72π4
27) 8π₯3π¦4 + 16π₯2π¦3 + 4π₯π¦ 28) 24π4 + 12π3 + 6π2
29) 40Ξ± +20Ξ±Ξ² +10Ξ±Ξ²Β΅ 30) 9π2 + 18π
23
Section 6.3: The Difference of Two Squares (DOTS)
βI know it happens to everyone, but it never happened to me.β
Factor Completely:
1) π₯2 β 9 30) 36 π4β10 β 49
2) π₯2 β 36 31) π16π36 β36
3) π¦2 β 64 32) 4π₯2 β 9
4) π¦2 β 81 33) π4 β π4
5) 100 β π§2 34) (ππ )2 β 25
6) 25 β π2 35) π6β4 β 25
7) π2 + 16
8) π₯2 + 49
9) π2 β 10
10) π₯2 β 19
11) π₯4 β 1
12) π₯4 β 16
13) π¦8 β 144
14) 16π₯16β 49
15) 36π¦16β 121
16) π10 β π2
17) 16π₯16β 49
18) π2β π2
19) π6β4 βπ2
20) 1 β 4π₯2
21) 9 β 25π¦2
22) 16π₯2 β 49π¦2
23) 81π₯2 β 4π§2
24) 64π¦2 β 121π€2
25) π§4 β π¦2
26) 36π2π 4 β π‘2
27) 25π6π8 β 4
28) π₯36 β 4
24
Section 6.4: Combination of Greatest Common Monomial Factor and Difference of Two Squares
βMy grandfather said that this reminded him of his grandfather, now this
Reminds me of my grandfatherβ
Factor Completely:
1) 2π₯2 β 2 2) 3π¦2 β 27 3) 6π§ β 24π§3
4) 4π2 β 36 5) 2 β 8π2
6) 27β 75π2
7) 4π2 β 100 8) 10π₯4 β 10 9) 2π¦4 β 32 10) 8π§2 β 18 11) 72π2 β 8 12) 36π36 β 36 13) 16β 16π16
14) π4π 2 β π 2
15) π₯4 β 121π₯2
16) 3π¦3 β 75π¦ 17) 4π2π2 β π4
18) 6πππ2 β 24ππ 19) 18π₯2π¦3π§4 β 2π₯2π¦3π§2
20) 4π₯4π¦2 β 36π¦2
21) π5 β π3
22) 4π4 β π2
23) 25π₯4 β 4π₯2
24) 5π₯π¦2 β 125π₯ 25) 9π₯2 β 81 26) 4π₯3 β 16π₯ 27) π3 β 144π 28) 7π3 β 63π 29) 9π3π3 β 36ππ 30) 5 β 5π2
25
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Section 6.5: Trinomial With Leading Coefficient One ( x2 + ππ₯ + π) Factor Completely:
1) π₯2 + 3π₯ + 2
2) π₯2 β 3π₯ + 2
3) π₯2 + π₯ β 2
4) π₯2 β π₯ β 2
5) π¦2 + 5π¦ + 6
6) π¦2 + π¦ β 6
7) π¦2 β π¦ β 6
8) π¦2 β 5π¦ β 6
9) π§2 + 4π§ + 3
10) π§2 β 4π§ + 3
11) π§2 β 2π§ β 3
12) π§2 + 2π§ β 3
13) π2 +10a +24
14) π2 + 7π β 30
15) π2 β 6π + 8
16) π2 β 13π + 30
17) π 2 β 9π + 18
18) π‘2 β 4π‘ β 12
19) π€2 + 2π€ β 15
20) π2 β 11π + 28
) π₯2 β 2π₯ β 48
) π¦2 β 3π¦ β 54
) π§2 β 17π§ + 30
) π2 + 4π β 60
) π2 + 10π + 25
) π2 + 5π β 14
) π2 β 16π + 28
) π₯2 β 8π₯ β 20
) π¦2 + 5π¦ β 66
) π£2 + 6π£ β 55
) π’2 + 6π’ + 5
) π2 β 3π β 10
) π2 β 6π β 16
) π2 + 7π β 18
) π2 + 9π + 20
) π₯2 β 5π₯ β 36
) π¦2 + 14π¦ + 40
) π§2 + 6π§ + 5
) π2 + 14π + 48
) π2 + π β 72
26
Section 6.6: Trinomials with Leading Coefficients Other Than One
(ππ₯2 + ππ₯ + π) Factor Completely:
1) 2π₯2 + 5π₯ + 2 2) 2π₯2 β 5π₯ + 2 3) 2π₯2 β 3π₯ β 2 4) 2π₯2 + 3π₯ β 2 5) 3π¦2 β 7π¦ + 2 6) 3π¦2 + 7π¦ + 2 7) 3π¦2 β 5π¦ + 2 8) 3π¦2 + 5π¦ + 2 9) 5π§2 β 14π§ β 3
10) 5π§2 + 14π§ β 3
11) 5π§2 β 16π§ + 3
12) 5π§2 + 16π§ + 3
13) 3π2 β π β 2
14) 5π2 + 2π β 3
15) 3π2 β 7π + 4
16) 4π2 + 5π β 6
17) 4π2 + 10π β 24
18) 6π2 + 11π + 5
19) 6π 2 β 7π β 5
20) 8π‘2 β 6π‘ β 9
21) 8π’2 + 18π’ + 9
22) 9π£2 + 12π£ + 4
23) 9π€2 + 6π€ β 8
24) 10π₯2 β 19π₯ + 6
25) 12π¦2 + 8π¦ β 15
26) 14π§2 β 19π§ β 3
27) 15π2 + π β 6
28) 16π2 β 4π + 25
29) 16π2 + 8π + 1
30) 20π2 + 3π β 2
27
Section 6.7: Combination of Greatest Common Monomial. Factor and Trinomial
Factor Completely:
1) 4π₯2 β 4π₯ β 8
2) 2π₯3 + 4π₯2 + 2π₯
3) π§5 β 2π§4 β 3π§3
4) 5π2 β 40π β 45
5) 6π2 β 6π β 120
6) 2π2 β 6π β 20
7) 4π2 + 20π + 24
8) 3π3 β 3π2 β 18π
9) 5π 4 β 30π 3 β 35π 2
10) 2π‘3 + 26π‘2 β 60π‘
11) 4π’4 + 12π’3 + 8π’2
12) 4π£2 + 16π£ + 12
13) 6π₯2 + 30π₯ + 36
14) 8π€3 + 24π€2 + 16π€
15) 8π¦2 + 32π¦ + 24
16) 10π§2 + 70π§ β 300
17) 11π2 β 66π+88
18) 12π4 + 24π3 β 180π2
19) 15π2 β 45π β 150
20) 20π2 β 160π β 400
28
Section 6.8: Factoring by Grouping
βThe person who does the work is the only one who learns.β
Factor by Grouping
1) π₯(π₯ + 4) + 2(π₯ + 4)
3) π§(π§ + 9) β 7(π§ + 9)
5) 4π(π + 7) + 3(π + 7)
7) 7π(π β 8) β 5(π β 8)
9) π₯3 β 6π₯2 + 4π₯ β 24
11) π§3 β 4π§2 + 3π§ β 12
13) 2π₯2 + 4π₯ + 3π₯ + 6
15) 4π§2 β π§ β 8π§ + 2
17) 3π₯3 + 9π₯2π¦ β 4π₯ β 12π¦
19) 5π2π+15π2π β 6π β 18π
21) 63π’2π£ + 18π’2π€ + 35π£π₯ + 10π€π₯
2) π¦(π¦ β 6) + 5(π¦ β 6)
4) π(π β 3) β 8(π β 3)
6) 5π(π + 11) β 8(π + 11)
8) 12π(π β 1) β 17(π β 1)
10) π¦3 + 5π¦2 β 6π¦ β 30
12) π4 + 3π3 β 6π β 18
14) 2π¦2 β 3π¦ β 14π¦ + 21
16) π2 β 6π β 6π + 36
18) 4π₯π¦2 + 20π¦2π§ + 3π₯ + 15π§
20) 21ππ β 35ππ‘ + 12π π£ β 20π‘π£
22) 12π₯π¦ + 15π₯π§ + 20π₯π + 25π§π
29
Section 6.9: Review of All Factoring
Factor Completely: 1) 36βπ₯2
2) π₯2 β 7π₯ β 60 3) 28π2π3 β 35π2π2 + 7ππ 4) 3π¦2 β 12π¦ β 15 5) π₯4 β 1 6) 8π¦3π§3 β 32π¦π§ 7) 16π16 β 25 8) 3π2 β 5π β 2 9) 4π₯π¦π§ + 5π₯π¦ + 6π¦π§ 10) 9 β 25π2
11) π4π2 β π2
12) π2 β 4π + 3 13) π2 + 7π β 30 14) 9π 2 + 12π + 4 15) π‘2 + 12π‘ + 10 16) 9π’3 β 27π’2 β 90π’ 17) 15π£3 + 12π£2
18) 12π₯4π¦ β 8π₯3π¦ β 4π₯2π¦ + 4π₯π¦ 19) 4π₯2 β 4π₯ 20) 42 β 13π₯ + π₯2
21) 20π₯2π¦ β 25π₯π¦2 + 30π₯π¦ 22) 6π§2 β 7π§ β 5 23) π₯4 β 16 24) 8π¦2 β 18 25) 2π§2 β 7π§ β 15 26) 2π3 β 20π2 + 48π 27) 21 + 10π + π2
28) 40π3π2 β 20π2π + 10ππ2 + 5ππ 29) π₯2 β 25 30) 6π₯2 β 5π₯ β 4 31) 6π₯2 + 6π₯ β 5π₯ β 5 32) 4π¦2 + 16π¦ + 3π¦ + 12 33) 6π₯π§2 + 9π¦π§2 + 10π₯ + 15π¦ 34) 15π2π β 20π2π β 21π + 28π
30
Section 7.1: Is a Given Number a Solution to an Equation
βIn mathematics, we never understand things, but just get used to them.β
Determine whether or not the given number is a solution to the equation 1) π₯ = β1, and 15π₯ + 20 = 5
2) π¦ = β2, and 12 β 5π¦ = 2
3) π§ = 3, and 10 β 4π§ = 1 β π§
4) π = 4, and 8 β 4π + 2 = 3(2 β π)
5) π = 2, and 3(π β 5) + 6 = 2(π + 1)
6) π = β3, and π β 4(π + 5) = 4 β 2(1 β π)
7) π = β1, and π2 β 4π + 3 = 0
8) π = 6, and π2 + 10π + 24 = 0
9) π = β2, and π 2 = π + 2
10) π‘ = 2, and (π‘ β 2)(π‘ + 5) = 0
11) π€ = β3, and (π€ β 3)(π€ + 1) = 0
12) π₯ = 8, and (π₯ β 8)(π₯ + 4)(π₯ + 3)(π₯ + 2) = 0
13) π¦ = β1, and 2π¦3 β 4π¦2 + π¦ β 9 = 0
14) π§ = 1, and 2π§4 β 3π§3 + 4π§2 + 5π§ β 6 = 0
15) π = β7, and 9 β (π β 3) + 4π = 5π β 5(π + 2) + 6
16) π = 4, and 3(2π β 3) β 4(3π β 1) + 5 = 6(4 β 2π)
17) π = 1, and 9(2π + 3) β 8π = 5(3π + 1) + 16
18) π₯ = β3, and 2(1 β 2π₯) + 3(4π₯ + 5) = 4 β (1 β 4π₯)
31
Section 7.2: Linear (First Degree) Equations
βTouching the dull formulas with his wand, he turned them into poetry.β
Solve each equation: 1) 4π₯ = 36 2) 6π₯ = 72 3) β5π₯ = 20 4) β8π₯ = 48 5) β2π¦ + 1 = 15 6) 2π¦ + 4π¦ = 54 7) π¦ + 3π¦ + 5π¦ = 81 8) 4π¦ + 6π¦ + 2π¦ = 60 9) 3π§ + 4 = 31 10) 5π§ β 9 = 41 11) 2π + 7 = 3π β 9 12) 3π + 7 = 8π β 18 13) 5π + 11 = 7π β 3 14) 2π₯ + 3π₯ β 9 = 4π₯ + 5π₯ + 3 15) 5 + 3(π₯ + 1) = 8 16) 2(π + 4) = 3(π β 5) 17) β4(π + 3) = 5(π + 3) 18) 8(π + 4) = 5π β 28 19) 20 β 5π = β2(π β 1) 20) 10 β 9π = 8 β 3(π + 2) 21) π β 6(π β 4) = 3 β (3 β 7π) 22) 5 β 2(π + 3) + 20 = 5(2π β 1) 23) 3π β 5(1 β π ) = 3(π + 1) + 2 24) 6 β 2(π‘ β 5) = 12 + 3(2π‘ β 4) 25) 3(π‘ β 2) β 2 = 5(π‘ + 3) β 7(π‘ β 1) 26) (π’ β 3) β 2(π’ β 4) = 5(π’ β 6) β π’ 27) 2(π£ β 3) β (π£ β 4) = 4(3 β π£) 28) 2(π₯ + 1) + 4(2π₯ β 3) = 3 β (4π₯ + 1) 29) 40π₯ + 60(π₯ β 1) = 40 + 20(π₯ β 1) 30) 5 β (π¦ β 3) + 3π¦ = 4π¦ β 5(π¦ + 3) β 1 31) 5 β 2(π§ + 3) + 20 = 5(2π§ β 1) 32) 2(3π₯ + 1) + 4(5π₯ + 2) + 6(7π₯ + 9) = 64 33) 20(2π§ + 3) + 40(3π§ + 4) + 60(8π§ + 1) = 920 34) 2(π§ + 3) + 5(π§ + 4) = 6(2π§ + 9) + 8(3π§ β 4)
32
Section 7.3: Decimal Equations
βWhatever we invent in mathematics seems independent of our inventing.β βMathematics lies in an enchanted world, somewhere between reality and
imagination.β
Solve for each variable: 1) . 4π₯ = .8 2) . 6π₯ = 1.8 3) 2.4π₯ + 1.2π₯ = 7.2 4) . 3π¦ + 2 = 1.2 5) 2.4π¦ + 3 = 7.8 6) π¦ + .4 = 5 7) . 02π§ = 4 8) . 4π§ = .16 9) . 02π§ = .7π§ + .68 10) 7.4 = 1.8π β 16 11) . 04π β 3.2 = .2π 12) . 3π β 1.2 = .1π + .4 13) . 48 β .09π = 1.88 14) 1.5π + 2 = 1.7π β 4.6 15) . 5(5π β 1) β .2(π + 4) = 1 16) . 25(3π β 4) β .2 = .25π + .3 17) 1.6π + 1.51 = .09π 18) . 48 β .07π = .89π 19) 1.2π β 4 = .4π + 28 20) 3.5π β 1.5 = 3.14π + 2.10 21) 3.6π‘ + 1.73 = 2.86π‘ β .49 22) . 17π€ + 1.56 = .14π€ + 1.2 23) π€ + 0.24 = 1.6π€ β 0.84 24) 0.02π₯ + 0.4π₯ + 2.1 = 0 25) . 08(π₯ + 1) β .4(π₯ + 2) = .24 26) . 04π₯(π₯ + 2) + .2(π₯ + 4) = .16 27) . 002π¦ = 6 28) . 004π¦ + 3.2 = .02π¦ 29) . 006π§ + .04π§ + .2π§ = .496 30) π₯ + .3π₯ + .03π₯ + .003π₯ = 3.9
33
Section 7.4: Fractional Equations with Monomials in the Numerator or Binomials in the Numerator
βOn earth, the broken arcs; in heaven, a perfect round.β
Solve each equation
π₯ 7π₯β
5π₯ π₯ 5 1) = 4 18) = +
5 6 8 3 24
3π₯ 1 3 7 27 2) = 9 19) + + =
7 π₯ 2π₯ 8π₯ 64
β5π₯ 1 5 4 3) = 10 20) + + = 1
8 4π₯ 12π₯ 3π₯
3 2π₯ π₯β1 4) π¦ β
14
= 21) = 2 3 4
1 π¦ 2 3 5) + = 2 22) =
9 27 π₯β4 π₯β5
π¦ π¦ 3 3π₯ π₯+1 6) + = 23) =
2 4 8 5 2
1 5 π₯ π₯β2 β
1 7) = 24) + = 6
π§ 3π§ 6 3 5
1 1 2 π₯+2 π₯β1 8) + = 25) + = 5
π 15 3 3 6
1 4 3 π₯β1 2(π₯+2) β
4 9) + = 26) + = 9
5π₯ 2 5 π₯ 9 3
π₯ 7π₯ 11 π₯+1 4π₯β1 2 10) + = 27) + =
3 12 6 5 15 15
2π 7 2(π₯β5) 3π₯ π₯+5 β
1 11) =b + 28) + =
3 2 6 6 5 10
34
12) π 3π 16
+ = 3 5 5
29) π₯ 2(π₯β1)
β π₯+5
+ = 1 5 9 10
13) π
β 5π β1
= 2 9 6
30) π₯ π₯ π₯ π₯
+ + + = 15 2 4 8 16
14) π‘ 2π‘ 23
+ = 9 5 15
15) π₯ π₯ π₯
+ + =12 12 6 4
16) 12 3 9
+ = π¦ 2π¦ 4
17) 1 3 2
+ = 2π§ 8 π§
35
Section 7.5: Literal Equations
βIβd rather listen to a tree than to a mathematician.β
Solve for the indicated variable:
1) Solve for r: π = ππ‘
2) Solve for l: π΄ = ππ€
3) Solve for I: π = πΌπ
4) Solve for r: πΆ = 2ππ
5) Solve for h: π΄ = 1
πβ 2
6) Solve for h: π = ππ€β
7) Solve for R: ππ = ππ π
8) Solve for r: πΌ = πππ‘
1 9) Solve form: πΎ = ππ£2
2
10) Solve for r: π = π£2βπ
11) Solve for t: π£2 = π£1 + ππ‘
12) Solve for a: π = 2π + 2π
13) Solve for b: π = π + π + π
14) Solve for C: πΉ = 9
πΆ + 32 5
15) Solve for l: π = 2πππ + 2ππ2
36
2π 5 16) Solve for w: π = 28) Solve for F: πΆ = (πΉ β 32)
π€ 9
πΊπ1π2 17) Solve for π1: πΉ = 29) Solve for z: 2π₯ + 3π¦ + 4π§ = 10
π2
18) Solve for π1: π΄ = β(π1 + π2) 30) Solve for y: π΄π₯ + π΅π¦ = πΆ
19) Solve for r: π΄ = π(1 + ππ‘)
ππ2β 20) Solve for h: π =
3
21) Solve for x: π¦2 = 4ππ₯
22) Solve for t: π = πΎ + ππ‘
ππ£2
23) Solve for r: πΉ = ππ
1 1 1 24) Solve for R: = +
π π 1 π 2
π1π1 π2π2 25) Solve for π2: =
π1 π2
26) Solve for y: 2π₯ + 3π¦ = 6
27) Solve for y: 4π₯ β 5π¦ = 20
37
Section 7.6: Quadratic (Second Degree) Equations
Solve for each variable: 1) (π₯ β 2)(π₯ β 5) = 0 36) 50π₯2 β 25π₯ = 0 2) (2π₯ + 1)(π₯ β 3) = 0 37) 36π¦2 = 24π¦ 3) π¦(π¦ β 6) = 0 38) 4π§2 + 12π§ + 8 = 0 4) π¦(10 β π¦) = 0 39) 6π2 + 11π + 5 = 0 5) 4π§(π§ β 9) = 0 40) π2 + 10π + 25 = 0 6) 8π§(π§ + 11) = 0 7) (π + 8)(π β 8) = 0 8) (6 β π)(6 + π) = 0 9) π₯2 β π₯ β 12 = 0 10) π₯2 + 2π₯ β 8 = 0 11) π₯2 β 8π₯ β 20 = 0 12) π₯2 + 2π₯ β 15 = 0 13) 2π¦2 + 11π¦ + 5 = 0 14) 6π¦2 + 13π¦ + 6 = 0 15) 4π¦2 + 5π¦ β 6 = 0 16) 10π¦2 β 19π¦ + 6 = 0 17) 4π§2 β 4π§ β 8 = 0 18) 6π§2 β 6π§ β 120 = 0 19) π§5 β 2π§4 β 3π§3 = 0 20) 5π§4 β 30π§3 β 35π§2 = 0 21) π2 β 4π = 12 22) π2 + 4π = 60 23) π2 = π + 2 24) π2 = π + 6 25) π2 + 8 = 6π 26) π 2 β 48 = 2π 27) π‘2 β 4π‘ = 0 28) 6π’2 β 3π’ = 0 29) π£2 β 81 = 0 30) 144 β π€2 = 0 31) (π₯ + 1)(π₯ + 2) = 0 32) (2π₯ + 3)((π₯ + 4) = 0 33) (π₯ + 5)(π₯ β 4) + 14 = 0 34) (π₯ + 6)(π₯ β 2) β 9 = 0 35) π¦2 β 4π¦ + 4 = 0
38
Section 7.7 Linear Inequalities
Solve the following inequalities and sketch the solution on the number line. 1) 2π₯ > 10 2) 4π₯ < β12 3) 5π₯ β₯ 20 4) 8π₯ β€ 32 5) β3π₯ > 15 6) β6π₯ β€ 12 7) β10π₯ β₯ 20 8) β15π₯ β€ 30 9) 8π₯ β₯ β16 10) 9π₯ β€ β18 11) β12π₯ > β24 12) β5π₯ < β15
π₯ 13) > 3
5
14) π₯
β€ 5 9
15) βπ₯
β₯ 4 7
16) βπ₯
< β2 3
17) 2π₯
β₯ 8 3
18) 4π₯
β€ 12 5
β3π₯ 19) > 9
7
20) β4π₯
< β 8 11
21) π₯ + 4 < 6
22) π₯ β 5 > β3
39
23) π₯ + 9 β€ 2
24) π₯ β 7 β₯ β4
25) 4π₯ β 6 β₯ 10
26) 5π₯ + 10 < 15
27) 8π₯ β 7 > 9
28) 6π₯ + 8 β€ β4
29) 3π₯ β€ 5π₯ + 8
30) 7π₯ β 10 < 2π₯
31) 4π₯ + 8 > 12π₯
32) 6π₯ + 10 β₯ 8π₯
33) 2π₯ β 5 > 3π₯ + 10
34) 4 β 5π₯ β€ π₯ β 2
35) 3π₯ + 4 β₯ 5π₯ β 6
36) 5π₯ + 4 < 9π₯ β 8
37) 5π₯ β (π₯ + 4) > 12
38) 3(π₯ + 2) β€ 3 β 2(π₯ + 3)
39) 6 β 3(π₯ + 1) β₯ 6
40) 4 β 5(π₯ + 2) < 1 β 2(π₯ β 1)
40
Section 8.1: Simplifying Fractions (Rational Expressions)β Monomial/Monomial; Polynomial/Monomial; Polynomial/Polynomial
βThere is a thin line that separates a numerator and denominator.β
Simplify:
12 1)
4
β9 2)
36
π₯3
3) π₯
β24π¦5
4) 3π¦2
β18π₯2π¦ 5)
β9π₯π¦
48π4π3
6) β16π2π2
β13π4π2
7) 39π6π3
β30π4π 8)
β6π3π
β66π3π7π9
9) 11π2π3π4
β45π6π 3π‘2
10) β9π2π π‘4
41
8π₯3β4π₯2+2π₯ 11)
2π₯
8π¦π§2β4π¦2π§+2π¦π§ 12)
β2π¦π§
12π3π2β6π2π+3ππ 13)
3ππ
24π¦12+12π¦6β6π¦3
14) β6π¦3
25π25+16π16β9π9
15) βπ6
12π7β9π5β3π4
16) β3π4
π₯β3 17)
3βπ₯
7βπ¦ 18)
π¦β7
4π₯β4 19)
π₯2β1
π¦2β9 20)
3π¦β9
6π§2β12π§ 21)
π§2β4
π2βπβ6 22)
π2β9
π2βπβ12 23)
π2β16
42
6π2β6 24)
π3βπ
3π2β3π 25)
π2+5πβ6
π 2βπ β2 26)
π 2β2π
π‘2βπ‘β12 27)
π‘2+5π‘+6
π₯2β9π₯+20 28)
π₯2β3π₯β10
π§2β6π§+8 29)
2π§β8
9π₯ 30)
π§2β6π§+8
72 π₯4π¦6π§3
31) 12 π₯2π¦3π§4
6π¦2+12π¦ 32)
π¦2+4π¦+4
43
Section 8.2: Multiplying Fractions (Rational Expressions)
βThe winners joke and the losers say deal.β
Multiply:
4β
12 1)
6 8
β3β
14 2)
7 6
β25 β
β4 3)
2 5
2 β
3β
4β
5β
6 4)
3 4 5 6 7
3 5) 14 β
7
6) 2π₯2
β π¦
π¦ 4π₯
7) π3
β π2
π4 π2
8) β4π2
β β5π2
15π 12π
9) 30π2
β 6π2
18π 5π
10) β6πππ
β 10π₯π¦2
5π₯2π¦ 3πππ2
11) β21π₯2π¦
β β16π§2
8π§ 3π₯π¦
44
β2π₯ β
6π§ β
14π 12)
3π¦ 7π 18π₯
β7π₯ β
β9π¦ β
2π¦ 13)
3π¦2 14π₯2 π₯
14) β44π3
β β10π2
β β2π2
5ππ 4π 11π
15) 3πππ
β β8π¦2π§
β 6π₯2
4π₯π¦ 9ππ 15π
(β3π) 16) 25π2π β
10π2
17) β7π2π π‘3
β β12π 3
β β4π3
6π 2 14ππ‘2 π‘
18) 24π2π3
β 10ππ
β βπ3
25π 8ππ2 9π2
β6π₯4π¦7π§9
β 30π₯2π¦3 39π¦2
19) β 5π₯2π¦ 13π₯π¦2 36π₯π¦π§4
20) β2πππ2
β β5π3
β β6π4
3π3π 4ππ2 35π2
π₯+1 π₯2β9 21) β
2π₯β6 π₯2+4π₯+3
4π¦3β4π¦ 9 22) β
27 π¦2β1
45
Section 8.3: Dividing Fractions (Rational Expressions)
βI was so much older then; Iβm younger than that now.β
Divide:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
2 4 Γ·
3 9
β5 10 Γ·
6 3
β9 β3 Γ·
11 22
π₯2 π₯2
Γ· π¦ π¦
π4 π2
π3 Γ· π
5π π4
Γ· 7π 21π2
4π₯π¦2 4π₯π¦2
3ππ2 Γ· 6ππ
9ππ2 12π4π3
10ππ 2 Γ· ππ
27ππ2 3ππ Γ·
10ππ 5ππ
60π 2π‘ 10π π‘ Γ·
7π’π£ π’2
8π2π β24ππ Γ·
27ππ 9
46
β15π₯2π¦2 β5π₯π¦2
12) Γ· 36ππ 12ππ
π5π4π2 π2π3π4
13) Γ· ππ π4π
β24π6π8π10 β12π4π6
14) Γ· 5ππ2 ππ
4π₯2π¦3π§4 β2π₯π¦2π§4
15) Γ· 5πππ 15ππ
25π₯2π¦ β5π₯π¦ 16)
21π4π3 Γ· π3π2
β72π8π 7 9π4π 5
17) Γ· 5ππ 10π2π2
14π₯2π¦π§3 β28π₯π¦π§ 18)
11π2π3π4 Γ· 22ππ2π3
5π₯ 19) 25π₯π¦ Γ·
3π¦
10π2π 20) Γ· 5πππ
7π₯π¦
β36π₯π¦4 β9π₯π¦ 21) Γ·
11ππ 22π
15ππ 2 5ππ 22) Γ·
7ππ 14
β16ππ β4ππ2
23) Γ· 9π2π 3ππ
2 4 24) Γ·
3π₯ 6π₯
47
π₯+1 π₯2β1 25) Γ·
π₯ π₯2
3π₯β21 π₯β7 26) Γ·
5 15
π₯π¦2 π₯π¦ 27) Γ·
π₯2βπ₯β12 π₯2β9
6π2
β 3ππ 10π3π2
28) Γ· 7π3 5 14π
48
Section 8.4: Adding or Subtracting Fractions (Rational Expressions)
βThree out of two people have trouble with fractions.β
Combine:
2 3 4 2 1) + 2) +
5 5 3 3
3 7 β
1 β
2β
1 3) 4)
10 10 9 9 9
1 1 1 1 5) + 6) +
2 3 2 4
1 1 1 1 5 7) + + 8) +
2 4 8 4 6
5 7 3π₯ π₯ 9) + 10) +
6 12 4 12
8π¦ 4π¦ 5 β
7 11) + 12)
3 7 6π§ 4π§ 2 4
β3
β5
13) 14) π π π2 π
6 7 1 β
5 15) 16)
π₯ 3π₯ 3π2 + 6π
4 5 1 1 17) + 18) +
9π¦ 12π¦ 2 3
3 5 β
2β
1 β
3 19) 20)
π₯2 π₯ 2π₯ 6π 8π
9 1 1 β
5 21) β 22)
3π 9π2 12π 6π
49
11 5 π₯ 2π₯β
3π₯ 23) + 24) +
8π 6π 2 10 5 2
π 2πβ
5π πβ
5πβ
π 25) + 26)
10 15 3 3 12 6
7 1 8 1 β
3 27) 28) + β
5π‘ π‘2 π€ 5π€ 10π€
π£β
π£ 5 3 29) β1 30) + β 3
10 5 12π 4π
7 2 3π¦ 2 β
3π¦ β
1 31) + 32) +
10π₯ 15 5π₯ 4 5π¦ 2
50
Section 9.1: The Pythagorean Theorem
βEuclid alone has looked on beauty bare.β βGeometry is the art of reasoning well from ill-drawn figures.β
Given right triangle ABC with leg AC (also called leg b), leg BC (also called leg a) And hypotenuse AB (also called side c), find the missing side in each problem.
A
b c
C B a
1) Leg AC =3, Leg BC =4, find hypotenuse AB
2) Leg AC =5, Leg BC =12, find hypotenuse AB
3) Leg a=8, leg b=15, find hypotenuse c
4) Leg a=7, leg b=24, find hypotenuse c
5) Leg a=24, hypotenuse c=26, find leg b
6) Leg a=30, hypotenuse c=34, find leg b
7) Leg BC=40, hypotenuse AB=41, find leg AC
8) Leg BC=48, hypotenuse AB=50, find leg AC
9) Hypotenuse c=37, leg b=35, find leg a
10) Hypotenuse c=61, leg b=60, find leg a
51
11) Side AC=6, side BC=8, find hypotenuse AB
12) Side AC=9, side BC=12, find hypotenuse AB
13) Side a=1, side b=1, find hypotenuse c
14) Side a=3, side b=3, find hypotenuse c
15) Side AC=2, side BC=3, find hypotenuse AB
16) Side AC=4, side BC=5 find hypotenuse AB
17) Hypotenuse AB=6, leg AC=4, find leg BC
18) Hypotenuse AB=6, leg AC=5, find leg BC
19) Hypotenuse c=6, leg a=4, find leg b
20) Hypotenuse c=6, leg a=3, find leg b
21) Side b=4, side a=8, find hypotenuse c
22) Side a=2, side b=6, find hypotenuse c
23) Side a=2, hypotenuse c=6, find side b
24) Side a=5, hypotenuse c=10, find side b
25) Side AC=5, side BC=10, find hypotenuse AB
26) Side AC= 3, side BC=6, find hypotenuse AB
52
Section 9.2: Simplifying Radicals--Numerical; Variable
βWhether you think you can or you think you canβtβyouβre right.β
Simplify:
1) β25 2) β49 3) ββ144 4) ββ81 5) β12 6) β20 7) ββ32 8) ββ50 9) 2β48 10) 4β27 11) β6β8 12) β10β24 13) 5β80 14) β15β50 15) βπ₯6 16) βπ¦8
17) ββπ§10 18) ββπ12
19) βπ7 20) βπ13
21) βπ9 22) βπ25
23) ββπ16 24) βπ 36
25) 2π₯βπ₯3 26) 3π¦βπ¦5
27) β4π§βπ§7 28) β9π2βπ11
29) βπ₯2π¦4 30) βπ3π4
31) ββπ10π12 32) βπ2π 15
33) β12π₯2 34) β75π¦3
35) β27π4 36) β72π5
37) β8π₯2π¦7 38) β50π4π8
39) β54π3π11 40) β200π9π 19
41) 4β32π₯2π¦3π§4 42) 6β72π4π5π6
43) β10β18π16π 9π‘25 44) β14π₯β27π₯6π¦10π§12
45) β2πβ144π14π20π40 46) β5β80π’7π£3π€13
47) 7π₯π¦β20π₯2π¦5π§8 48) 12π2ππ3β50π2π7π4
49) β24ππ 2π‘5β18π2π 10π‘12 50) β20π’5π£9π€β72π’7π£3π€5
53
Section 9.3 Multiplying Radicals
βYou have a greater claim to the throne if you pull the sword from the stone
than if you say βIβm the kingβ β.
Multiply:
1) β3 Γ β5 3) β6 β β30 5) β5 Γ β40 7) 2β3 β 4β5 9) 6β5 Γ 3β15 11) 2π₯β3π₯3 β 4π₯β6π₯2
13) 5π§2β2π§ β 6π§4β8π§ 15) β4πβ3π β 8πβ3π
17) 2π₯β5π₯π¦π§ β 6π₯β10π₯2π¦3π§5
19) β8π₯β2π₯π¦π§ β 4π₯β6π₯2π¦π§3
21) 3(β7 β 8) 23) β5(β5 + 2) 25) β5(β15 β β5) 27) 3β2(4β2 β 6β12)
29) (β3 + 5)(β6 + 4) 31) (β7 + 5)(β7 β 5) 33) (10 + β6)(10 β β6) 35) (β5 + 4)2
37) (β7 β 5)2
39) (β2 + β3)2
2) β2 Γ β7 4) β2 β β8 6) β3 Γ β30 8) 6β7 β 8β11 10) β2β3 Γ 4β27 12) β3π¦β6π¦7 β 2π¦β5π¦4
14) 4πβ10π8 β 3πβ2π5
16) 5π₯π¦β3π₯2π¦ β 4π₯π¦β6π₯π¦
18) β2ππ2β8ππ5π β 4π2πβ6πππ2
20) β7ππ π‘2β6π3π π‘5 β 10β3ππ π‘5
22) 4(β6 β 5) 24) β3(β3 + 6) 26) β2(5β2 β 4β8) 28) 5β3(16β6 β 3β27)
30) (β2 + 3)(β6 β 8) 32) (β3 + 6)(β3 β 6) 34) (12 β β8)(12 + β8) 36) (β6 + 3)2
38) (β8 β 9)2
40) (β5 + β6)2
54
Section 9.4: Dividing Radicals
βI wish I didnβt know now what I didnβt know then.β
Divide:
1) β25
2) β49
36 81
β200 β48 3) 4)
β2 β3
β40 β200 5) 6)
β5 β10
14β15 9β24 7) 8)
2β5 3β12
6β20 24β12 9) 10)
3β2 6β3
β18π₯3 β24π3π 6 11) 12)
β2π₯ β2ππ 2
β32π5π7 β40π₯3π¦4 13) 14)
β8π3π5 β10π₯π¦2
β5 β8 15) 16)
β3 β7
β12 β9 17) 18)
β50 β6
19) β7
20) β10
3 7
55
21) β π₯
24 22) β
π¦2
5
23 βπ₯8
π¦6 24) β20π§5
25
5 25)
β5
7 26)
β7
4β8ββ12 27)
2β3
2β6β15β12 28)
5β2
8β15β2β5 29)
4β3
7β3β4β20 30)
15β5
4π₯5β2π₯7β6π₯3β45π₯3 31)
8π₯2β5π₯
2π¦3β3π¦5β4π¦2β27π¦7 32)
8π¦β3π¦
4β8β5β6 33)
2β2
27β5β3β10 34)
9β2
24βπ₯9 35)
β6π₯
50βπ¦21 36)
β5π¦
10 37)
β2 38) β
10
2
4 39)
β5+1
6 40)
β2β1
56
Section 9.5 Combining Radicals
βExperience is what you get when you donβt get what you want.β
Combine:
1) 2β7 + 3β7
3) 8β5 + 4β5 β 6β5
5) 10β3π₯ β 7β3π₯ β 5β3π₯
7) 7β5πππ β 4β5πππ β 6β5πππ
9) 4β3 β 2β27
11) 5β12 β 6β75
13) 9β20 β 2β45
15) 6β32 + 8β200 β 5β98
17) 2β80 + 11β125 β 5β45
19) 8β20π₯3 β 6π₯β5π₯
21) 5βπ2π + 6β4π2π β 5πβ16π
23) 2β12π₯π¦ + 4β27π₯π¦ β 5β75π₯π¦
25) 5β8 + 3β50 β 9β125
2) 11β3 β 5β3
4) 12β2 β 9β2 β 10β2
6) 14β6π₯ + 2β6π₯ β 3β6π₯
8) 17β7ππ π‘ + 4β7ππ π‘ β 8β7ππ π‘
10) 7β2 + 4β8
12) 8β72 β 10β18
14) 3β50 β 5β8
16) 5β48 β 9β75 β 3β300
18) 3β8 + 6β72 β 9β50
20) 6β12π₯5 β 8π₯2β3π₯
22) 7βππ 3 + 9π βππ β 2π β16ππ
24) 3β5π¦π§ β 4β45π¦π§ β 9β20π¦π§
26) 2β18 + 4β27 β 6β12
57
Section 10.1: The Rectangular Coordinate System
Define the following terms 1) Horizontal Axis (X-Axis) 2) Vertical Axis (Y-Axis) 3) Origin 4) Quadrant I, Quadrant II, Quadrant III, Quadrant IV 5) Ordered Pair 6) Abscissa, Ordinate
Plot (Graph) the following points (Ordered Pairs) 7) A(2,4); B(1,6) 8) C(6,-1); D(8,-3) 9) E(-2,5); F(-5,3) 10) G(-1,-6); H(-3,-7) 11) K(0,4); L(0,-3) 12) M(1,0); N(-5,0)
13) P(4 ,1); Q(2, -2
1)
2 2
14) R(-1 1, 7); S(3
2 , 6)
3 3
Plot the following points on one coordinate axis 15) A(0,1); B(1,3); C(2,5); D(3,7); E(-1,-1)
Plot the following points on one coordinate axis 16) R(0,-1); S(1,2); T(2,5); U(3,8); V(-1,-4)
Find the coordinates for each of the following points (ordered pairs) 17) A; B 18) C; D 19) E; F 20) G; H 21) J; K
58
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
G
F
J
D
K
H
A
BC
59
Section 10.2: Graphing By Use of the Table Method
Complete the given table and sketch the graph for:
1) π¦ = π₯ + 3 π₯ π¦ 0 2
4
2) π¦ = π₯ β 2
3) π¦ = 3π₯ + 2
4) π¦ = 4π₯ β 3
5) π₯ + π¦ = 6
π₯ π¦ 0 2 4
π₯ π¦ 0 1 2
π₯ π¦ 0 1 2
π₯ π¦ 0 2 4
π₯ π¦ 60
6) π₯ β π¦ = 4
7) 2π₯ + 3π¦ = 12
8) 3π₯ β 4π¦ = 12
9) 6π₯ + 3π¦ β 18 =0
10) 2π₯ β 5π¦ β 10 = 0
11) π¦ = 1
π₯ + 3 4
1 12) π¦ =
6π₯ β 5
0 2
4
π₯ π¦ 0 3 6
π₯ π¦ 0 4 6
π₯ π¦ -2 0 3
π₯ π¦ -5 0
5
π₯ π¦ 0 4
8
π₯ π¦ 0 6
12
61
Section 10.3: Graphing by Use of the Intercepts Method
Identify a) the Y-intercept and b) the X-intercept in each of the following equations. 1) 2π₯ + 3π¦ = 6 2) 3π₯ + 4π¦ = 12 3) 5π₯ β 6π¦ = 30 4) 7π₯ β 3π¦ = 21 5) 3π₯ + 2π¦ = 7 6) 4π₯ + 6π¦ = 9 7) 5π₯ β 3π¦ + 11 = 0 8) 8π₯ β 5π¦ β 12 = 0 9) π¦ = 2π₯ β 3 10) π¦ = 5π₯ β 8
Complete the given table and sketch the graph for each of the following equations.
11) π₯ + π¦ = 5 19) 2π₯ β 3π¦ = 9 π₯ π¦
0 0
2
π₯ π¦ 0
0
2
12) π₯ β π¦ = 7 20) 3π₯ β 4π¦ = 8
π₯ π¦
0 0
4
π₯ π¦ 0
0
2
13) 2π₯ + 5π¦ = 10
π₯ π¦
0 0
3
62
14) 9π₯ + 2π¦ = 18 21) π¦ = 2π₯ + 3
π₯ π¦
0 0
1
π₯ π¦ 0
0
-1
15) 3π₯ β 4π¦ = 12 22) 3π₯ β 4π¦ = 8
π₯ π¦
0 0
2
π₯ π¦ 0
0
2
16) 3π₯ β 5π¦ = 15
17) π₯ + 2π¦ = 7
π₯ π¦
0 0
3
π₯ π¦
0 0
5
18) 2π₯ + 3π¦ = 11
π₯ π¦
0 0
3
63
8
6
4
2
0
-8 -6 -4 -2 -2
0 2 4 6 8
-4
-6
-8
-10
Section 10.4: Graphing by Use of the Slope-Y Intercept Method
Find the slope, y-intercept, and sketch the graph for each of the following equations. 1) π¦ = 2π₯ + 3 2) π¦ = 4π₯ + 6 3) π¦ = 5π₯ β 2 4) π¦ = 7π₯ β 9
1 β1 5) π¦ =
2π₯ + 3 6) π¦ =
4 π₯ + 6
7) π¦ = 2π₯ 8) π¦ = π₯ 9) π¦ = 4 10) π¦ = β2 11) 2π₯ + 3π¦ = 6 12) 4π₯ + 5π¦ = 20 13) 3π₯ β 4π¦ = 12 14) 5π₯ β 2π¦ = 10 15) π₯ + 4π¦ = 24 16) 3π₯ β 5π¦ = 15
Write the equation of the line passing through the two given points in each of the following problems. 17) A(1,2); B(3,4) 18) C(2,1); D(-1,4) 19) E(3,5); F(6,8) 20) G(4,7); H(8,9) 21) K(-3,1); L(2,6) 22) M(-2,4); N(6,2) 23) P(-1,-3); Q(-2,-6) 24) R(-4,-7); S(-2,-5)
Write the equation of the vertical line passing through the given points. 25) S(2,5) 26) T(4,-6) 27) U(-8,9) 28) V(-7,-3)
Write the equation of the horizontal line passing through the given points. 29) S(2,5) 30) T(4,-6) 31) U(-8,9) 32) V(-7,-3)
Given the following graphs, write the equation of the line in slope and y-intercept form (π¦ = ππ₯ + π) 33) 10
-10 10
64
34)
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
35)
65
-8 -6 -4
8
6
4
2
0
-2 -2
-4
-6
-8
-10
0 2 4 6 8
-8 -6 -4
8
6
4
2
0
-2 -2
-4
-6
-8
-10
0 2 4 6 8
36) 10
-10 10
-10 10
37) 10
66
Section 11.1: Solving a System of 2 Equations in 2 Unknowns Graphically
Solve the following systems of 2 equations in 2 unknowns graphically. If a system does not have a unique solution, identify the system as either dependent or inconsistent.
1) π₯ + π¦ = 4 2) π₯ + π¦ = 6 π₯ β π¦ = 2 2π₯ β π¦ = 6
3) 3π₯ + π¦ = 6 4) π₯ β π¦ = 1 π₯ + π¦ = 4 3π₯ β 2π¦ = 6
5) π¦ = 2π₯ 6) π¦ = 3π₯ π¦ = π₯ + 2 π¦ = 2π₯ + 1
7) π₯ + π¦ = 4 8) π₯ β 3π¦ = β2 π¦ = 2π₯ β 5 π¦ = π₯ + 2
9) 2π₯ + 3π¦ = 8 10) 3π₯ + 4π¦ = 10 4π₯ + π¦ = 6 4π₯ β 3π¦ = 5
11) 2π₯ + 3π¦ = 6 12) 4π₯ + 3π¦ = 10 π¦ = 4 π₯ = 1
13) 3π₯ + 5π¦ = 15 14) 5π₯ β 2π¦ = 10 π₯ + 2π¦ = 5 2π₯ β π¦ = 3
15) 5π₯ β 3π¦ = 15 16) 3π₯ β 4π¦ = 12 2π₯ β π¦ = 7 π₯ β 2π¦ = 2
17) π₯ + π¦ = 2 18) π₯ β π¦ = 3 π₯ + π¦ = 4 π¦ = π₯ β 4
19) π₯ + π¦ = 3 20) 3π₯ + 4π¦ = 12 2π₯ + 2π¦ = 6 6π₯ + 8π¦ = 24
67
Section 11.2: Solving a System of 2 Equations in 2 Unknowns by Use of the Substitution Method
Solve the following systems of equations by use of the Substitution Method.
1) 3π₯ + π¦ = 15 2) 5π₯ β 2π¦ = 7 π¦ = 3 π¦ = 4
3) 2π₯ β 3π¦ = 4 4) 6π₯ + 4π¦ = 14 π₯ = 5 π₯ = 1
5) π¦ = π₯ + 2 6) π¦ = π₯ β 3 π₯ + π¦ = 4 π₯ + 2π¦ = 6
7) π₯ = π¦ + 5 8) π₯ = π¦ β 1 2π₯ β π¦ = 12 3π₯ + 2π¦ = 17
9) π₯ + π¦ = 6 10) π₯ + π¦ = 5 4π₯ β 3π¦ = 10 5π₯ β 4π¦ = β2
11) 2π₯ + π¦ = 5 12) 3π₯ + π¦ = 10 6π₯ β π¦ = 3 7π₯ β 3π¦ = 2
13) 5π₯ + 3π¦ = 6 14) 3π₯ β 2π¦ = 2 4π₯ + 2π¦ = 6 2π₯ + 4π¦ = 28
15) π₯ β π¦ = β1 16) 2π₯ β π¦ = 2 8π₯ β 5π¦ = 13 9π₯ β 10π¦ = 5
17) 3π β 2π = 4 18) 5π β 3π = 1 11π β 20π = 2 10π β 7π = β1
19) 4π + π β 10 = 0 20) 2π β π β 5 = 0 5π + 2π β 14 = 0 4π β π β 1 = 0
68
Section 11.3: Solving a System of 2 Equations in 2 Unknowns by Use of the Elimination Method (Adding or Subtracting)
Solve the following systems of equations by the Elimination Method.
1) π₯ + π¦ = 3 π₯ β π¦ = 1
3) βπ₯ + 2π¦ = 2 π₯ + 3π¦ = 13
5) 2π₯ + 3π¦ = 3 4π₯ β 3π¦ = 15
7) π₯ + 2π¦ = 1 2π₯ + 3π¦ = β1
9) 4π₯ + 3π¦ = β2 3π₯ + π¦ = β4
11) 5π₯ + 4π¦ = 13 6π₯ β 3π¦ = 39
13) 2π + 3π = 6 3π β 2π = β17
15) 7π β 4π = 22 5π + 6π = 60
17) 6π’ + 7π£ = 3 4π’ β 3π£ = 25
19) π₯ β 3π¦ + 15 = 0
5π₯ β 2π¦ + 23 = 0
2) 2π₯ + π¦ = 4 3π₯ β π¦ = 1
4) βπ₯ + π¦ = 1 π₯ + 2π¦ = 11
6) 3π₯ + 4π¦ = 5 π₯ β 4π¦ = β9
8) π₯ β 3π¦ = 7 3π₯ + 2π¦ = 10
10) 5π₯ β 4π¦ = β3 6π₯ β 2π¦ = β12
12) 6π₯ + 3π¦ = 6 9π₯ β 5π¦ = β48
14) 3π β 4π = 23 2π + 5π = 0
16) 8π + 3π = β2 3π β 4π = β11
18) 2π€ β 5π§ = 16 3π€ + 4π§ = 7
π₯ β
π¦ 20) =1
2 6 π₯
β π¦
= 5 3 4
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Section 12.1: Ratio and Proportion
βI was good at math before they decided to mix the alphabet into it.β
1) A carβs gas tank holds 20 gallons of fuel. If it can travel 500 miles on one tank full of gas, how many gallons of gas will it need to travel 120 miles?
2) A car uses 4 gallons of gas to go 120 miles. At that rate of usage, how many
gallons will it take to travel: a) 480 miles; b) 360 miles; c) 200 miles?
3) If 64 cups of coffee can be made using 1 pound of coffee, about how many
pounds of coffee will be needed to make 200 cups?
4) If a town in New York State assesses a real estate tax of $2.50 per $100 of
assessed valuation, what will the real estate tax be on a house with assessed
valuation of $11,000?
5) If 4 ounces of oil is needed to be added to 1 quart of gasoline to operate a chain
saw, how much oil will be needed for 1 gallon of gasoline (4 quarts)?
6) If a six foot tall man cast a four foot long shadow, how long will the shadow be
for a person of height of five feet?
7) If 6 ears of corn sells for $2.00, what will the cost be for a) 18 ears; b) 27 ears; c)
36 ears?
8) If a machine can produce 400 items in 8 hours, how long will it take to make: a)
1000 items; b) 1600 items; c) 2600 items?
9) If on average, 6 students receive Aβs in a class of 30 students, how many calculus
students are expected to get Aβs in a semester if there are: a) 450 students; b) 750
students; c) 1000 students?
10) If an instructor can grade 30 exam papers in one hour, how long will it take her
to grade: a) 100 papers; b) 150 papers; c) 175 papers?
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Section 12.2: Percent Problems
βI meant what I said and I said what I meantβ¦An Elephantβs faithful one hundred percent!β
1) What is 6% of 200?
2) What is 9% of 400?
3) Five percent of what number is 300?
4) Eight percent of what number is 500?
5) Ninety percent of the students in a statistics class of 30 students passed their
first exam, if so, how many students actually passed this examination?
6) If 20% of a calculus class of 40 students received Aβs as a final grade, how many
students received Aβs?
7) Sixty percent of eligible voters voted in the most recent election. If there were
120 million eligible voters, how many voted?
8) A car can travel 500 miles on a full tank of gas. If it has travelled 350 miles,
a) what percent of its fuel has been used and b) what percent remains?
9) A new laptop computer has a list price of $800. If the price were reduced by
20%, a) what is the amount of the reduction and b) what is the new price?
10) A new car has a Manufacturerβs Suggested Retail Price (MSRP) of $25,000. If the cat can be purchased for 12% less than the MSRP, what is the new selling price?
11) A restaurant bill comes to $45.00. If one wants to leave a 20% tip, a) what is the
amount of the tip and b) what is the total cost of the meal?
12) An employee starts working at a new job at the rate of $12.00 per hour. If after 3
months, his salary is increase by 10%, what will his new salary be?
13) A light bulb manufacturing company found that 40 out of 1000 bulbs that it
produced were defective. What was the percent of defective bulbs produced?
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14) Thirty years ago, the cost to cross the Henry Hudson Bridge was 10 cents. Today
the cost is $2.50. A) What is the percent increase over the past 30 years? B) At this
rate, what will the cost to cross the bridge be 30 years from now?
15) A new car cost $20,000. If it depreciates (reduces in value) by 9% once you
drive it out of the showroom, how much will it be worth?
16) In a class of 25 students, 6 received Bβs. What percent of the class received
grades of B?
17) New York State has a 7% sales tax on many items. If a television costs $500, a)
what is the tax on the TV and b) what is the total cost?
18) If you were to borrow $2,000 for one year at the interest rate of 5% per year
(compounded annually), a) how much interest would you owe and b) what amount
would you need to repay after one year?
19) At one college 40% of the entering class takes pre-calculus. If there are 2000
students in the entering class, a) how many take pre-calculus and b) how many
students do not take pre-calculus?
20) If a real estate broker receives a 5% commission on the sale of a house and she
receives a $20,000 commission, what is the selling price of the home?
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Section 12.3: Number Problems
βSmart people learn from their mistakes. Very smart people learn from other peoplesβ mistakes.β
1) The sum of a given number and 7 is 25, what is the number?
2) The sum of a given number and 14 is 35, what is the number?
3) The sum of twice a number and 6 is 18, what is the number?
4) The sum of two consecutive numbers is 25, what are the two numbers?
5) The sum of two consecutive numbers is three times the original number, what is
the original number?
6) The sum of two numbers is 36. If one is twice the other, what are the two
numbers?
7) If three more than four times a number is 39. Find the number.
8) If four less than twice a number is 12, find the number.
9) If eight less than three times a number is 22, find the number.
10) If two times a number is increased by four, the result is 20. Find the original
number.
11) If a number is subtracted from 30, the result is 10. Find the number.
12) Six times a number is reduced by 5, the result is 67. Find the number.
13) If three times a number is increased by ten, the result is three less than four
times the original number. Find the original number.
14) Four times a number is decreased by five and equals seven more than twice the
original number. Find the original number.
15) A fifteen foot long board is cut into two pieces. Find the length of each piece if:
a) one is twice the length of the other; b) one is three feet longer than the other; and
c) one is three feet longer than twice the length of the other.
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Section 12.4: Distance = Rate Γ Time
βHappiness is not about what the world gives youβhappiness is what you think
about what the world gives you.β
1) Two cars start from the same point and travel in opposite directions. If they each
travel at 60 miles per hour, how far apart will they be after 4 hours of travel?
2) Two small airplanes start at the same point and travel in exactly opposite
directions. If one travels at 200 miles per hour and the other travels at 150 miles per
hour, how far apart will they be after 3 hours of travel?
3) A man on a bicycle travels 48 miles in 6 hours. What was the average speed for
this trip?
4) If an airplane makes a 2500 mile trip in 5 hours, what was the average rate of
speed for the plane?
5) If an airplane travels 400 miles per hour for 3 Β½ hours, how far did the plane go?
6) If it takes a jogger 12 minutes to run a mile, at that rate of speed, how many
miles will she cover in 2 hours?
7) A hiker can walk 32 miles in an 8 hour day. What is her average rate of speed for
this hike?
8) If a hiker can travel 12 miles per day over flat terrain, 8 miles per day over steep
terrain, and 24 miles per day in a canoe, how many miles will the hiker have
travelled in 6 days if there were 1 day of canoeing, 2 days of steep hiking, and 3 days
of flat hiking?
9) A truck driver leaves a factory at 8:00 am and travels till noon and then stops for
1 hour for a lunch break. If the truck driver then drives till 4:00 pm and averaged 55
miles per hour while driving, how far did he go?
10) The Pacific Coast Trail (PCT) is 2659 miles long. If a hiker covers the trail in 5
months (150 days), a) how many miles per day did she cover and b) if she hikes for
10 hours a day, what was her average rate per hour?
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11) The Pacific Coast Trail (PCT) is 4279 kilometers long. If a hiker covers the trail
in 5 months (150 days), a) how many kilometers per day did he cover and b) if he
hikes for 10 hours a day, what was her average rate per hour?
12) The Appalachian Trail (AT) is 2181 miles long. If a hiker covers the trail in 4
months (120 days), a) how many miles per day did she cover and b) if she hikes for
10 hours a day, what was her average rate per hour?
13) The Appalachian (AT) is 3510 kilometers long. If a hiker covers the trail in 4
months (120 days), a) how many miles per day did he cover and b) if he hikes for 10
hours a day, what was her average rate per hour?
14) Two cars start from the same point and travel in exactly opposite directions. If one travels at 65 miles per hour and the other at 50 miles per hour, after 6 hours: a) how far apart will the cars be; b) if they each can go 30 miles on 1 gallon of gasoline, how much gas will be used ; and c) if gas costs $2.50 per gallon, how much must each car pay?
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Answers to Odd Numbered Problems
Section 1.1: Nomenclature-Terms, Constants, Monomials, Binomials, Trinomials, Absolute Values
1) Coefficent: 8; Variable: x 3) Coefficent: 7; Variables: x and y 5) Coefficent: 12; Variables: x, y and z 7) Monomial 9) Monomial 11) Trinomial 13) Terms: 4π₯2, 8π₯, β10 15) Terms: β6π₯3, 7π₯2, β3π₯, 4 17) Variables: 10π₯2, β7π₯ ; Constant: 2 19) Variables: β7π₯4, 3π₯2, β4π₯ ; Constant: 17 21) Degree: 2; Number of Terms: 3; Constant: -7 23) Degree: 16; Number of Terms: 4; Constant: -21 25) Degree: 7; Number of Terms: 5; Constant: none 27) 4 29) 24
3 31)
8 33) 11
Section 1.2: Exponents 1) 36 = 729 3) 16 5) -16 7) π₯8
9) π11
11) π10π15
13) β20π₯7π¦5
15) 21π3π5
17) 2π₯6
19) 8π§16
21) 16π8π12
23) π2
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25) π4
1 27)
π6
1 29)
π2 1
31) π‘8
33 ) π€13
1 35)
π¦11
16 37)
9 9
39) 4 1
41) 16
43) β1
16 45) b
128 47)
π18
49) 625π₯8
125π₯6
51) π¦12
53) β16π5
Section 1.3: Scientific Notation 1) 2.4 X 103
3) 2.8 X 102
5) 3 X 10β2
7) 4.5 X 10β7
9) 480,000 11) 8,140,000,000 13) .094 15) .356 17) 6 X 107 = 60,000,000 19) 4.212 X 1017 = 421,200,000,000,000,000 21) 8 X 10β8 = .00000008 23) 2.34 X 10β7 = .000000234 25) 3 X 102 = 300
77
27) 3 X 10β8 = .00000003 29) 2 31) 1.02 X 106 = 1,020,000 33) 1 X 10β6 = .000001
Section 2.1: Operations Involving Addition, Subtraction, Multiplication, or Division 1) 30 3) -21 5) -10 7) -45 9) 21 11) 33 13) -36 15) -160 17) 6 19) -8 21) -2 23) 2 25) 9 27) -2 29) 0 31) 10 33) 5 35) 7 37) -21 39) 12 41) 20 43) -7 45) -5 47) 18 49) -5 51) 0 53) 0 55) 22 57) 32 59) -3 61) 4
78
Section 2.2: Order of Operations 1) 34 3) 0 5) 5 7) 20 9) 40 11) 6 13) 19 15) 28 17) 8 19) 17 21) 2 23) 1 25) 2 27) -5 29) -1 31) -17 33) 79 35) 26 37) 10 39) -96
Section 3.1: Evaluating Algebraic Expressions 1) 12 3) 18 5) 48 7) 36 9) 1 11) 8 13) -9 15) -28 17) 9 19) 12 21) -8 23) 4
79
25) -4 27) 12
Section 3.2: Functional Notation 1) 17 3) 1 5) -12 7) -39 9) -34 11) 4 13) 9 15) 6 17) 3 19) 0 21) 14 23) 2 25) 4a+4h+3 27) π2 + 2πβ + β2 + 4π + 4β + 6 29) β4π2 + 2π β 6
Section 4.1: Multiply Monomial β Monomial
1) π₯5
3) 20π8
5) β18π 12
7) 24π3π3
9) β10π6π 4
11) 9π₯8π¦10
13) 64π₯4
15) 54π₯4π¦4
17) 90π7π 12
19) 60π₯3π¦5π§9
21) 24π5π14
23) β24π₯5π¦9π§8
25) β30π11π12π15
27) β18π5π8
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29) 180π₯7π¦4
31) β20π₯6
Section 4.2: Multiply Monomial β BinomialβThe Distributive Law 1) 16π₯ + 48 3) 18π₯3 + 42π₯ 5) β2π¦4 β 6π¦2
7) β27π§4 + 45π§3
9) 2π4 + 5π3
11) 44π2 β 8π3
13) β2π3 + 20π2 β 18π 15) 3π₯4π¦2 β 4π₯3π¦3 + 9π₯2π¦ 17) β2π₯4 β 10π₯3 + 12π₯2
19) β20π7π 5 β 24π6π 4 β 16π6π 3 + 4π4π 3
Section 4.3: Binomial β Binomial 1) π₯2 + 4π₯ + 3 3) π₯2 + 2π₯ β 35 5) π¦2 + 10π¦ + 16 7) βπ¦2 + π¦ + 6 9) 2π2 + 21π + 27 11) 12π2 + 11π β 15 13) 12π2 β 28π + 15 15) 15π₯2 β 25π₯ + 10 17) 18π₯4 β 69π₯3 β 12π₯2
19) π₯2 β 6π₯ + 9 21) 9π₯2 β 42π₯ + 49 23) π₯2 β 16 25) 36βπ¦2
27) 16π₯2 β 25 29) 20π₯3 + 20π₯2 + 5π₯
Section 4.4: Multiply Binomial β Trinomial or Trinomial β Trinomial 1) π₯3 + 3π₯2 + 5π₯ + 3 3) π₯3 β 5π₯2 + 14π₯ β 24
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5) 2π¦3 β 3π¦2 β 32π¦ + 48 7) 5π§3 β 5π§2 + 34π§ β 48 9) 4π§4 + 36π§3 + 6π§2 + 51π§-27 11) π3 + 9π2 + 15π + 7 13) π3 + 3π2 + 3π + 1 15) 8π3 + 12π2 + 6π + 1 17) π₯4 + 5π₯3 + 8π₯2 + 5π₯ + 7 19) 2π₯4 + 18π₯3 + 16π₯2 β 120π₯
Section 5.1: Combining Like Terms 1) β11π₯ 3) 8π₯2π¦ 5) 2π₯ + 16π¦ 7) 23π β 32 9) 7ππ 2 + 10 11) 2π₯2π¦ + 10π₯π¦2 β 8π₯π¦ 13) β11π3 β 24ππ β 7π 15) 4π₯2 + π₯ β 5 17) 3π₯3 β 15π₯2 + 5π₯ β 16 19) 3π₯2 + 19π₯ β 27 21) β6π3π + 9π2π2 β 12π2π3 + 5π2π 23) 9 β 14π₯ + 14π¦ 25) 4π₯2 β 5π₯ β 2 27) β34π3π 3 + 18π2π 2 + 10ππ 29) 16π2π β 9π2π2
31) 10π₯3π¦2 β 26π₯ β 2π₯3π¦ + 2π₯2
33) 0 35) β38π₯2 + 49π₯ β 45
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Section 6.1: Unique Prime Factorization of Integers 1) 22 β 3 3) 25 β 3 5) 2 β 32
7) 22 β 5 9) 23 β 32
11) 32 β 11 13) 24 β 3 15) 24 β 5 17) 22 β 33
19) 2 β 3 β 7 21) 33
23) 25 β 3 β 5 25) 2 β 53
27) 24 β 32 β 5 29) 23 β 32 β 5
Section 6.2: Greatest Common Monomial Factor (GCF)-Numerical or Algebraic 1) 2(π₯ β 2) 3)π§(5π§ + 8) 5) 4π¦(π¦2 + 5π¦ β 2) 7) 7π(π2 β 2π + 3) 9) 12π₯π¦3(3π₯ + 1) 11) 7π(2π2 β 4π + 5) 13) 5π3(4π7 + 2π2 β 1) 15) 9π₯2π¦2π§2(2π₯π¦2π§3 β 3π¦π§ + 1) 17) 11π2ππ(1 β 2π + 3ππ) 19) π¦π§(2π₯π§ β 3π₯ + 4) 21) 4(π₯3 + 2) 23) 15π₯π¦2(1 + 2π¦) 25) 2(3π₯2 + 8π¦2 + 13π§2) 27) 4π₯π¦(2π₯2π¦3 + 4π₯π¦2 + 1) 29) 10πΌ(4 + 2π½ + π)
83
Section 6.3: The Difference of Two Squares (DOTS) 1) (π₯ + 3)(π₯ β 3) 3) (π¦ + 8)(π¦ β 8) 5) (10 + π§)(10 β π§) 7) Not Factorable 9) Not Factorable 11) (π₯2 + 1)(π₯ + 1)(π₯ β 1) 13) (π¦4 + 12)(π¦4 β 12) 15) (6π¦8 + 11)(6π¦8 β 11) 17) (4π¦8 + 7)(4π¦8 β 7) 19) (π3β2 + π)(π3β2 β π) 21) (3 + 5π¦)(3 β 5π¦) 23) (9π₯ + 2π§)(9π₯ β 2π§) 25) (π₯2 + π¦)(π₯2 β π¦) 27) (5π3π4 + 2)(5π3π4 β 2) 29) (π3β2 + π)(π3β2 β π) 31) (π8π18 + 6)(π8π18 β 6) 33) (π2 + π2)(π + π)(π β π)
Section 6.4: Combination of Greatest Common Factor and Difference of Two Squares
1) 2(π₯2 β 1) = 2(π₯ + 1)(π₯ β 1) 3) 6π§(1 β 4π§2) = 6π§(1 + 2π§)(1 β 2π§) 5) 2(1 β 4π2) = 2(1 + 2π)(1 β 2π) 7) 4(π2 β 25) = 4(π + 5)(π β 5) 9) 2(π¦4 β 16) = 2(π¦2 + 4)(π¦2 β 4) = 2(π¦2 + 4)(π¦ + 2)(π¦ β 2) 11) 8(9π2 β 1) = 8(3π + 1)(3π β 1) 13) 16(1 β π16) = 16(1 + π8)(1 β π8) = 16(1 + π8)(1 + π4)(1 β π4)
= 16(1 + π8)(1 + π4)(1 + π2)(1 β π2) = 16(1 + π8)(1 + π4)(1 + π2)(1 + π)(1 β π)
15) π₯2(π₯2 β 121) = π₯2(π₯ + 11)(π₯ β 11) 17) π2(4π2 β π2) = π2(2π + π)(2π β π) 19) 2π₯2π¦3π§2(9π§2 β 1) = 2π₯2π¦3π§2(3π§ + 1)(3π§ β 1) 21) π3(π2 β 1) = π3(π + 1)(π β 1) 23) π₯2(25π₯2 β 4) = π₯2(5π₯ + 2)(5π₯ β 2) 25) 9(π₯2 β 9) = 9(π₯ + 3)(π₯ β 3)
84
27) π(π2 β 144) = π(π + 12)(π β 12) 29) 9ππ(π2π2 β 4) = 9ππ(ππ + 2)(ππ β 2)
Section 6.5: Trinomial With Leading Coefficient One (π₯2 + ππ₯ + π) 1) (π₯ + 2)(π₯ + 1) 3) (π₯ + 2)(π₯ β 1) 5) (π¦ + 3)(π¦ + 2) 7) (π¦ β 3)(π¦ + 2) 9) (π§ + 3)(π§ + 1) 11) (π§ β 3)(π§ + 1) 13) (π + 6)(π + 4) 15) (π β 4)(π β 2) 17) (π β 6)(π β 3) 19) (π€ β 5)(π€ + 3) 21) (π₯ β 8)(π₯ + 6) 23) (π§ β 15)(π§ β 2) 25) (π + 5)(π + 5) 27) (π β 14)(π β 2) 29) (π¦ + 11)(π¦ β 6) 31) (π’ + 5)(π’ + 1) 33) (π β 8)(π + 2) 35) (π + 5)(π + 4) 37) (π¦ + 10)(π¦ + 4) 39) (π + 8)(π + 6)
Section 6.6: Trinomial With Leading Coefficient Other Than One (ππ₯2 + ππ₯ + π)
1) (2π + 1)(π + 2) 3) (2π₯ + 1)(π₯ β 2) 5) (3π¦ β 1)(π¦ β 2) 7) (3π¦ β 2)(π¦ β 1) 9) (5π§ + 1)(π§ β 3) 11) (5π§ β 1)(π§ β 3) 13) (3π + 2)(π β 1) 15) (3π β 4)(π β 1)
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17) 2(2π2 + 5π β 12) = 2(2π β 3)(π + 4) 19) (3π β 5)(2π + 1) 21) (4π’ + 3)(2π’ + 3) 23) (3π€ + 4)(3π€ β 2) 25) (6π¦ β 5)(2π¦ + 3) 27) (5π β 3)(3π + 2) 29) (4π + 1)(4π + 1)
Section 6.7: Combination of Greatest Common Monomial Factor and Trinomial 1) 4(π₯2 β π₯ β 2) = 4(π₯ β 2)(π₯ + 1) 3) π§3(π§2 β 2π§ β 3) = π§3(π§ β 3)(π§ + 1) 5) 6(π2 β π β 20) = 6(π β 5)(π + 4) 7) 4(π2 + 5π + 6) = 4(π + 2)(π + 3) 9) 5π 2(π 2 β 6π β 7) = 5π 2(π β 7)(π + 1) 11) 4π’2(π’2 + 3π’ + 2) = 4π’2(π’ + 2)(π’ + 1) 13) 6(π₯2 + 5π₯ + 6) = 6(π₯ + 3)(π₯ + 2) 15) 8(π¦2 + 4π¦ + 3) = 8(π¦ + 3)(π¦ + 1) 17) 11(π2 β 6π + 8) = 11(π β 4)(π β 2) 19) 15(π2 β 3π β 10) = 15(π β 5)(π + 2)
Section 6.8: Factoring by Grouping 1) (π₯ + 4)(π₯ + 2) 3) (π§ + 9)(π§ β 7) 5) (π + 7)(4π + 3) 7) (π β 8)(7π β 5) 9) (π₯ β 6)(π₯2 + 4) 11) (π§ β 4)(π§2 + 3) 13) (π₯ + 2)(2π₯ + 3) 15) (4π§ β 1)(π§ β 2) 17) (π₯ + 3π¦)(3π₯2 β 4) 19) (π + 3π)(5π2 β 6) 21) (7π£ + 2π€)(9π’2 + 5π₯)
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Section 6.9: Review of all Factoring 1) (6 + π₯)(6 β π₯) 3) 7ππ(4ππ2 β 5ππ + 1) 5) (π₯2 + 1)(π₯2 β 1) = (π₯2 + 1)(π₯ + 1)(π₯ β 1) 7) (4π8 + 5)(4π8 β 5) 9) π¦(4π₯π§ + 5π₯ + 6π§) 11) π2(π4 β 1) = π2(π2 + 1)(π2 β 1) = π2(π2 + 1)(π + 1)(π β 1) 13) (π + 10)(π β 3) 15) Not Factorable 17) 3π£2(5π£ + 4) 19) 4π₯(π₯ β 1) 21) 5π₯π¦(4π₯ β 5π¦ + 6) 23) (π₯2 + 4)(π₯2 β 4) = (π₯2 + 4)(π₯ + 2)(π₯ β 2) 25) (2π§ + 3)(π§ β 5) 27) (7 + π)(3 + π) 29) (π₯ + 5)(π₯ β 5) 31) (6π₯ β 5)(π₯ + 1) 33) (2π₯ + 3π¦)(3π§2 + 5)
Section 7.1: Is a Given Number a Solution to an Equation 1) 5 = 5, Yes 3) β2 = β2, Yes 5) β3 β 6, No 7) 8 β 0, No 9) 4 β 0, No 11) 12 β 0, No 13) β16 β 0, No 15) 12 β β4, No 17) 37 β 36, No
Section 7.2: Linear (First Degree) Equations 1) π₯ = 9 3) π₯ = β4 5) π¦ = β7 7) π¦ = 9 9) π§ = 9
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11) π = 16 13) π = 7 15) π₯ = 0 17) π = 3 19) π = 6
21) π = 3β2 23) π = 7β2 25) π‘ = 6
27) π£ = 14β5 29) π₯ = 1 31) π§ = 2 33) π§ + 10
Section 7.3: Decimal Equations 1) π₯ = 2 3) π₯ = 2 5) π¦ = 2 7) π§ = 200 9) π§ = β1
11) π = 4β3 13) π = 140β9 15) π = 1 17) π = β1 19) π = 40 21) π‘ = β3
23) π€ = 9β5 25) π₯ = 2 27) π¦ = 3000 29) π§ = 2
Section 7.4: Fractional Equations with Monomials or Binomials in the Numerator 1) π₯ = 20 3) π₯ = β16 5) π¦ = 51
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7) π§ = 4 9) π₯ = β3 11) π = β5 13) π = 3 15) π₯ = 12 17) π§ = 4 19) π₯ = 8
21) π₯ = β3β5 23) π₯ = 5 25) π₯ = 9 27) π₯ = 0
29) π₯ = 155β29
Section 7.5: Literal Equations
1) π = πβπ‘ 3) πΌ = πβπ 5) β = 2π΄βπ 7) π = ππβππ 9) π = 2πβπ£2
(π£2 β π£1) 11) π = βπ 13) π = π β π β π
(π β 2ππ2) 15) π = β
2ππ (πΉπ2)
17) π1 = βπΊπ2 (πΌ β π)
19) π = βππ‘ π¦2
21) π₯ = β4π
23) π = ππ£2βπΉπ
π1π1π2 25) π£2 = βπ2π1 (4π₯ β 20)
27) π¦ = β5
(10 β 2π₯ β 3π¦) 29) π§ = β4
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Section 7.6: Quadratic (Second Degree) Equations 1) π₯ = 2, 5 3) π¦ = 0, 6 5) π§ = 0, 9 7) π = 8, β8 9) π₯ = 4, β3 11) π₯ = 10, β2
13) π¦ = β5, β1β2 15) π¦ = β2, 3β4 17) π§ = 2, β1 19) π§ = 3, β1, 0 21) π = 6, β2 23) π = 2, β1 25) π = 4, 2 27) π‘ = 4, 0 29) π£ = 9, β9 31) π₯ = β1, β2 33) π₯ = 2, β3 35) π¦ = 2 37) π₯ = 1, β1 39) π§ = β2, β1
Section 7.7: Linear Inequalities 1) π₯ > 5 3) π₯ β₯ 4 5) π₯ < β5 7) π₯ β€ β2 9) π₯ β₯ β2 11) π₯ < 2 13) π₯ > 15 15) π₯ β€ β28 17) π₯ β₯ 12 19) π₯ < β28 21) π₯ < 2 23) π₯ β€ β27
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25) π₯ β₯ 4 27) π₯ > 2 29) π₯ β₯ β4 31) π₯ < 1 33) π₯ < β15 35) π₯ β€ 5 37) π₯ > 4 39) π₯ β€ β1
Section 8.1: Simplifying Fractions (Rational Expressions) 1) 3 3) β π₯2
5) 2π₯ β1
7) 3π2π
9) β6ππ4π5
11) 4π₯2 β 2π₯ + 1 13) 4π2π β 2π + 1 15) β25π19 β 16π7 + 9π3
17) -1 4(π₯β1) 4
19) = (π₯+1)(π₯β1) (π₯+1)
6π§(π§β2) 6π§ 21) =
(π§+2)(π§β2) (π§+2) (πβ4)(π+3) (π+3)
23) = (π+4)(πβ4) (π+4)
3π(πβ1) 3π 25) =
(π+6)(πβ1) (π+6) (π‘β4)(π‘+3) (π‘β4)
27) = (π‘+3)(π‘+2) (π‘+2) (π§β4)(π§β2) (π§β2)
29) = 2(π§β4) 2
6π₯2π¦3
31) π§
Section 8.2: Multiplying Fractions (Rational Expressions) 1) 1 3) 10
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5) 6π
7) π2
9) 2ππ 11) 14π₯π§ 13) 3
β4π₯π¦ 15)
15 17) β4π4π 2
19) β3π₯2π¦8π§5
1 21)
2
Section 8.3: Dividing Fractions (Rational Expressions) 3
1) 2
3) 6
π2
5) π2 β8π¦
7) ππ₯
9π 9)
2 1
11) 9π
13) π3π3πr β6π₯π¦
15) π
17) β16π4π 2ππ 19) 15π¦2
8π¦3
21) π 4
23) 3ππ
π₯ 25)
π₯β1 π¦(π₯β3)
27) π₯β4
Section 8.4: Adding or Subtracting Fractions (Rational Expressions)
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1) 1 1
3) 5 5
5) 6 7
7) 8 17
9) 12
8π¦ 11)
21 (2πβ3π)
13) ππ
13 15)
3π₯ 31
17) 36π¦ (6β3π₯)
19) 2π₯2
(27πβ1) 21)
9π (33π +20)
23) 24π 2
β43π 25)
30 (7π‘β15)
27) 5π‘2
(3π£β10) 29)
10 (4π₯+15)
31) 30π₯
Section 9.1: The Pythagorean Theorem 1) AB=5 3) c=17 5) b=10 7) AC=9 9) a=12 11) AB=10
13) c=β2
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15) AB=β13 17) BC=2β5 19) b=2β5 21) c=4β3 23) b=4β2 25) AB=5β5
Section 9.2: Simplifying Radicals 1) 5 3) β12 5) 2β3 7) β4β3 9) 8β3 11) β12β2 13) 20β5 15) π₯3
17) β π§5
19) π3βπ 21) π4βπ 23) β π8
25) 2π₯2βπ₯ 27) β4π§4βπ§ 29) π₯π¦2
31) β π5π6
33) 2π₯β3 35) 3π2β3 37) 2π₯π¦3β2π¦
39) 3ππ5β6ππ 41) 16π₯π¦π§2β2π¦
43) β30π8π 4π‘12β2π π‘ 45) β24π8π10π20
47) 14π₯2π¦3π§4β5π¦ 49) β72π2π 7π‘11β2
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Section 9.3 Multiplying Radicals
1) β15 3) 6β5 5) 10β2 7) 8β15 9) 90β3 11) 24π₯4β2π₯ 13) 120π§7
15) β96π3
17) 60π₯3π¦2π§3
19) β64π₯3π¦π§2β3π₯ 21) 3β7 β 24 23) 5 + 2β3 25) 5β3 β 5 27) 24 β 36β6 29) 3β2 + 4β3 + 5β6 + 20 31) β23 33) 94
35) 21 + 8β5 37) 32 β 10β7 39) 5 + 2β6
Section 9.4: Dividing Radicals 5
1) 6
3) 10
5) 2β2 7) 7β3 9) 2β10 11) 3π₯ 13) 2ππ
β15 15)
3
95
5
β6 17)
β21 19)
3 β6π₯
21) 12
π₯4
23) π¦3
25) β5 27) 8β2 29) 20
31) 9π₯10β2π₯ 33) 20β6 35) 4π₯4β6 37) 5β2 39) β5 β 1
Section 9.5: Combining Radicals
1) 5β7 3) 2β5 5) β2β3π₯ 7) β3β5πππ 9) β2β3 11) β20β3 13) 12β5 15) 69β2 17) 48β5 19) 10π₯β5π₯ 21) β3πβπ 23) β9β3π₯π¦
25) 25β2 β 45β5
Section 10.1: The Rectangular Coordinate System
96
1) One of two perpendicular lines intersecting at the origin. It is also called the abscissa and indicates how far left or right of the origin the point is. 3) The point where the x-axis and y-axis meet indicated by (0, 0) 5) Two numbers usually written with parenthesis (a, b) where the first number indicates how far to the left or right the point is and the second number indicates how high up or down. 17) A (2, 6) and B (8, 3) 19) No point E and F (2, 0) 21) J (4, -4) and K (-8, -3)
Section 10.2 Graphing by Use of the Table Method 1) π¦ = π₯ + 3
π₯ π¦ 0 3 2 5
4 7
2) π¦ = π₯ β 2
3) π¦ = 3π₯ + 2
4) π¦ = 4π₯ β 3
π₯ π¦ 0 -2 2 0 4 2
π₯ π¦ 0 2 1 5 2 8
π₯ π¦ 0 -3 1 1
2 5
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5) π₯ + π¦ = 6 π₯ π¦ 0 6 2 4 4 2
6) π₯ β π¦ = 4
7) 2π₯ + 3π¦ = 12
8) 3π₯ β 4π¦ = 12
9) 6π₯ + 3π¦ β 18 =0
10) 2π₯ β 5π¦ β 10 = 0
π₯ π¦ 0 -4 2 -2
4 0
π₯ π¦ 0 4 3 2
6 0
π₯ π¦ 0 -3 4 0 6 1.5
π₯ π¦ -2 10 0 6 3 0
π₯ π¦ -5 -4 0 -2 5 0
π₯ π¦
98
1 11) π¦ =
4π₯ + 3 0 3
4 4
8 5
1 12) π¦ =
6π₯ β 5 π₯ π¦
0 -5 6 -4 12 -3
Section 10.3: Graphing by Use of the Intercepts Method 1) The x-intercept is (0, 2) and the y-intercept is (3, 0) 3) The x-intercept is (0, -5) and the y-intercept is (6, 0)
7 7 5) The x-intercept is (0, ) and the y-intercept is ( , 0)
2 3 β11 β11
7) The x-intercept is (0, ) and the y-intercept is ( , 0) 3 5
3 9) The x-intercept is (0, -3) and the y-intercept is ( , 0)
2
11) π₯ + π¦ = 5
π₯ π¦
0 5 5 0
2 3
13) 2π₯ + 5π¦ = 10
π₯ π¦
0 2 5 0
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3 4
5
15) 3π₯ β 4π¦ = 12
17) π₯ + 2π¦ = 7
π₯ π¦
0 -3 4 0
2 β3
2
π₯ π¦
0 7
2 7 0
5 1
π₯ π¦
0 -3 9
2 0
2 β5
3
19) 2π₯ β 3π¦ = 9
100
21) π¦ = 2π₯ + 3
π₯ π¦
0 3 β3
2 0
-1 1
Section 10.4 Graphing by Use of the Slope-Y Intercept Method 1) Slope is m=2 and the y-intercept is (0, 3) 3) Slope is m=5 and the y-intercept is (0, -2)
1 5) Slope is m= and the y-intercept is (0, 3)
2 7) Slope is m=2 and the y-intercept is (0, 0) 9) Slope is m=0 and the y-intercept is (0, 4)
β2 11) Slope is m= and the y-intercept is (0, 2)
3 3
13) Slope is m= and the y-intercept is (0, -3) 4 β1
15) Slope is m= and the y-intercept is (0, 6) 4
17) π¦ = π₯ + 1 19) π¦ = π₯ + 2 21) π¦ = π₯ + 4 23) π¦ = 3π₯ 25) π₯ = 2 27) π₯ = β8 29) π¦ = 5 31) π¦ = 8 33) π¦ = π₯ + 2
35) π¦ = β4
π₯ 5
π₯ 8 37) π¦ =
3+
3
Section 11.1: Solving a System of 2 Equations in 2 Unknowns Graphically 1) (3, 1) 3) (1, 3)
101
5) (2, 4) 7) (3, 1) 9) (1, 2) 11) (-3, 4) 13) (5, 0) 15) (6, 5) 17) Inconsistent 19) Dependent
Section 11.2: Solving a System of 2 Equations in 2 Unknowns by the Use of the Substitution Method
1) (4, 3) 3) (5, 2) 5) (1, 3) 7) (7, 2) 9) (4, 2) 11) (1, 3) 13) (3, -3) 15) (6, 7) 17) (2, 1) 19) (2, 2)
Section 11.3: Solving a System of 2 Equations in 2 Unknowns by the Use of the Elimination Method (Adding or Subtracting)
1) (2, 1) 3) (4, 3) 5) (3, -1) 7) (-5, 3) 9) (-2, 2) 11) (5, -3) 13) (-3, 4) 15) (6, 5) 17) (4, -3) 19) (-3, 4)
Section 12.1: Ratio and Proportion
102
1) 4.8 gallons 3) 3 and 1/8 pounds 5) 16 ounces 7) a) $6; b) $9; c) $12 9) a) 90 students; b) 150 students; c) 200 students
Section 12.2: Percent Problems 1) 12 3) 6000 5) 27 students 7) 72 million 9) a) $160 reduction; b) $640 new price 11) a) $9 tip; b) $54 13) 4% 15) $18,200 17) a) $35 sales tax; b) $535 19) a) 800 students; b) 1200 students
Section 12.3: Number Problems 1) 18 3) 6 5) 1 7) 9 9) 10 11) 20 13) 13 15) a) 5 feet and 10 feet; b) 6 feet and 9 feet; c) 4 feet and 11 feet
Section 12.4: Distance = Rate X Time 1) 480 miles 3) 8 miles per hour 5) 1400 miles 7) 4 miles per hour 9) 385 miles
103