Precalculus Practice Problems, Methods, and Solutions Second Edition 2024 by Mehdi Rahmani-Andebili PDF | Complete Problem Sets with Step-by-Step Worked-Out Solutions | Ideal for High School and College Students in Precalculus, Algebra, Trigonometry, and

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Contents




1 Problems: Real Number Systems, Exponents and Radicals,
and Absolute Values and Inequalities ......................................................................... 1
1.1 Real Number Systems.......................................................................................... 1
1.2 Exponents and Radicals ....................................................................................... 3
1.3 Absolute Values and Inequalities........................................................................ 11
Reference................................................................................................................... 15
2 Solutions to Problems: Real Number Systems, Exponents
and Radicals, and Absolute Values and Inequalities ................................................ 17
2.1 Real Number Systems........................................................................................ 17
2.2 Exponents and Radicals ..................................................................................... 19
2.3 Absolute Values and Inequalities........................................................................ 26
Reference................................................................................................................... 29
3 Problems: Systems of Equations ............................................................................... 31
Reference................................................................................................................... 40
4 Solutions to Problems: Systems of Equations ........................................................... 41
Reference................................................................................................................... 47
5 Problems: Quadratic Equations ................................................................................ 49
Reference................................................................................................................... 58
6 Solutions to Problems: Quadratic Equations............................................................. 59
Reference................................................................................................................... 69
7 Problems: Functions, Algebra of Functions, and Inverse Functions........................ 71
Reference................................................................................................................... 87
8 Solutions to Problems: Functions, Algebra of Functions,
and Inverse Functions ............................................................................................... 89
Reference................................................................................................................. 103
9 Problems: Factorization of Polynomials ................................................................. 105
Reference................................................................................................................. 113
10 Solutions to Problems: Factorization of Polynomials ............................................. 115
Reference................................................................................................................. 120
11 Problems: Trigonometric and Inverse Trigonometric Functions ........................... 121
Reference................................................................................................................. 130




ix

,x Contents

12 Solutions to Problems: Trigonometric and Inverse Trigonometric
Functions ................................................................................................................. 131
Reference................................................................................................................. 143
13 Problems: Arithmetic and Geometric Sequences .................................................... 145
Reference................................................................................................................. 155
14 Solutions to Problems: Arithmetic and Geometric Sequences ................................ 157
Reference................................................................................................................. 166

Index............................................................................................................................... 167

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Problems: Real Number Systems, Exponents and
Radicals, and Absolute Values
and Inequalities
1


Abstract
In this chapter, the basic and advanced problems of real number systems, exponents, radicals, absolute values, and inequalities
are presented. To help students study the chapter in the most efficient way, the problems are categorized into different levels
based on their difficulty (easy, normal, and hard) and calculation amounts (small, normal, and large). Moreover, the problems
are ordered from the easiest, with the smallest computations, to the most difficult, with the largest calculations.


1.1 Real Number Systems
1.1. Which one of the numbers below exists [1]?
Difficulty level ● Easy ○ Normal ○ Hard
Calculation amount ● Small ○ Normal ○ Large
1) The minimum integer number smaller than -1.
2) The minimum irrational number larger than -1.
3) The maximum integer number smaller than -1.
4) The maximum rational number smaller than -1.

1.2. As we know, ℝ is the set of real numbers, ℤ is the set of integer numbers, and ℕ is the set of natural numbers. Which one
of the choices is correct?
Difficulty level ● Easy ○ Normal ○ Hard
Calculation amount ● Small ○ Normal ○ Large
1) ℕ ⊂ ℤ ⊂ ℝ
2) ℝ ⊂ ℤ ⊂ ℕ
3) ℝ ⊂ ℕ ⊂ ℤ
4) ℤ ⊂ ℝ ⊂ ℕ

Exercise: Which one of the rational numbers below can be considered an integer number?
1
1)
2

1

4

3

Final answer: Choice (2).



Ⓒ The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 1
M. Rahmani-Andebili, Precalculus, https://doi.org/10.1007/978-3-031-49364-5_1

,2 1 Problems: Real Number Systems, Exponents and Radicals, and Absolute Values and Inequalities



Exercise: Which one of the choices below is not an irrational number?
1) e
2) π
3) 0.3


Final answer: Choice (3).


a c
1.3. If = , then which one of the choices is correct?
b d
Difficulty level ● Easy ○ Normal ○ Hard
Calculation
b a amount ● Small ○ Normal ○ Large
1) =
c b
b d
2) ≠
a c
a b
3) ≠
bc dd
4) =
a c

1.4. Which one of the choices has the greatest absolute value?
Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ● Small ○ Normal ○ Large
, ,
1) - 1 +, 2 - 3,
2) - 1 - 2 + 3
, ,
3) - 1 - 2 - 3
, ,
4) 1 + 2 - 3

Exercise: Which one of the choices has the greatest value?




Final answer: Choice (4).


,
1.5. Which one of the choices below might be a rational number if α and β2 are rational and irrational numbers,
respectively?
Difficulty level ○ Easy ○ Normal ● Hard
Calculation amount ● Small ○ Normal ○ Large
1) α + β4
2) α + β
β
3) ,
1+ α
4) α2 + β

,1.2 Exponents and Radicals 3

1.2 Exponents and Radicals
3
,
1.6. Calculate the value of x2 if x = 2 2.
Difficulty level ● Easy ○ Normal ○ Hard
Calculation
, amount ● Small ○ Normal ○ Large
1) ,2
2) 3 2
,
3) 3 4
4) 2

3
Exercise: Simplify the term 3.
4
1) 39
1
2) 39
1
3) 33
7
4) 39

Final answer: Choice (1).


-1
1.7. Calculate the value of 1
-1
2 .
Difficulty level ● Easy ○ Normal ○ Hard
Calculation amount ● Small ○ Normal ○ Large
,
1) - 1 + ,2
2) - 2 + 2
,
3) - 1 + 2
,
4) - 2 + 2


Exercise: Calculate the value of +1 1 .
1) 1




Final answer: Choice (1).



3 3 , 2
1.8. Simplify and calculate the final answer of (- x) + x2 + (- 2) if x > 0.
Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ● Small ○ Normal ○ Large
1) -2x - 2
2) -2
3) 2x + 2
4) 2

,4 1 Problems: Real Number Systems, Exponents and Radicals, and Absolute Values and Inequalities


,
(- x)3 + x2 if x < 0.
3
Exercise: Calculate the final answer of
1) 2x
2) -2x
3) 0
4) -x3 + x2

Final answer: Choice (2).



, 5
10 3
1.9. Calculate the final answer of - 36 .
Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ● Small ○ Normal ○ Large
1) -9
2) 9
3) -3
4) 3

1
9 3
Exercise: Calculate the value of 83 .




Final answer: Choice (4).


, ,
1.10. Calculate the value of 2 3 x+
3 4 x 4 if x < 0.
Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ● Small ○ Normal ○ Large
1) 3x
2) x
3) -x
4) -3x

1.11. In the equation below, determine the value of x.

-8
1
3x - 1 =
81

Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ● Small ○ Normal ○ Large
1) 24
2) 27
3) 31
4) 33

,1.2 Exponents and Radicals 5



Exercise: In the following equation, calculate the value of x.

1
5 =
25

1) 5
2) 4
3) 3
4) 2

Final answer: Choice (1).



1.12. Calculate the value of 22 + 1 for k = 3.
k


Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ● Small ○ Normal ○ Large
1) 33
2) 65
3) 129
4) 257

2
Exercise: Calculate the value of 22 .
1) 16
2) 4
3) 8
4) 32

Final answer: Choice (1).



1.13. Solve the equation below.
7x - 3
3x - 7
3 = 7
7 3
Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ● Small ○ Normal ○ Large
1) -1
2) 1
3
3)
7
7
4)
3

,6 1 Problems: Real Number Systems, Exponents and Radicals, and Absolute Values and Inequalities



Exercise: Solve the following equation.


a = b
b a

1) 1
2) 2
3) 3
4) 0

Final answer: Choice (4).



1.14. Calculate the value of the following term.

-3
8
× (0.8) 4 × 0.2
25

Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ○ Small ● Normal ○ Large
2
1)
5
2) 2
5
3)
2
4) 5


Exercise: Calculate the value of the term below.

2
8 × (0.8)
25

25
1) 4
2) 1
4
3)
25
4
4)
5

Final answer: Choice (3).



1.15. Calculate the value of the term below.

5
25
× 3 × (0.75) - 3
90 2

Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ○ Small ● Normal ○ Large

, 1.2 Exponents and Radicals 7

5
1)
2
10
2)
3
3) 5
15
4)
2

1.16. Calculate the value of the term below.
6
-6 4
16 × × (0.5) × 8 - 3
2

Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ○ Small ● Normal ○ Large
1
1)
8
2) 2
3) 8
4) 4

1.17. Calculate the value of the term below.

3 × (45)6
(15) 6× 37

Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ○ Small ● Normal ○ Large
1) 1
2) 3
3) 5
4) 15


Exercise: Calculate the value of the term below.

2 × (10)5
(5)6 × 25
2
1)
5
5
2)
2
3) 1
4) 10

Final answer: Choice (1).



1.18. Calculate the value of x in the following equation.

1 1
, 1
1 2 2

(16)
x 2
2= 3




Difficulty level ○ Easy ● Normal ○ Hard
Calculation amount ○ Small ● Normal ○ Large