,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
, Problems: Real Number Systems, Exponents
and Radicals, and Absolute Values 1
and Inequalities
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
2
2) -
1
3
3) -
4
4
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
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
, Problems: Real Number Systems, Exponents
and Radicals, and Absolute Values 1
and Inequalities
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
2
2) -
1
3
3) -
4
4
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
