Inequalities

An inequality is a comparison established between two real numbers "aa", "bb" using order relation symbols, which can be true or false.

D

Definitions:

concept

∀a;b;c∈R\forall a; b; c \in \mathbb{R}

I. a is positive↔a>0a \text{ is positive} \leftrightarrow a > 0
II. a is negative↔a<0a \text{ is negative} \leftrightarrow a < 0
III. a>b↔a−b>0a > b \leftrightarrow a - b > 0
IV. a<b↔a−b<0a < b \leftrightarrow a - b < 0
V. a≤b↔a<b∨a=ba \le b \leftrightarrow a < b \lor a = b
VI. a≥b↔a>b∨a=ba \ge b \leftrightarrow a > b \lor a = b
VII. a<b<c↔a<b∧b<ca < b < c \leftrightarrow a < b \land b < c
VIII. a>b↔b<aa > b \leftrightarrow b < a

Law of Trichotomy

∀a∈R\forall a \in \mathbb{R} one, and only one, of the following three relations can hold:

a>0∨a=0∨a<0\boxed{a > 0 \quad \lor \quad a = 0 \quad \lor \quad a < 0}

Law of Closure

∀a;b∈R\forall a; b \in \mathbb{R} it holds that: (a+b)∈R+∧(a⋅b)∈R+(a + b) \in \mathbb{R}^+ \quad \land \quad (a \cdot b) \in \mathbb{R}^+

Theorem

∀a;b∈R\forall a; b \in \mathbb{R} we have: a>b∨a=b∨a<ba > b \quad \lor \quad a = b \quad \lor \quad a < b

Fundamental Theorems of Inequalities ∀a;b;c;d∈R\forall a; b; c; d \in \mathbb{R}

  1. a<b↔a+c<b+ca < b \leftrightarrow a + c < b + c
  2. If a>b∧b>c→a>ca > b \land b > c \rightarrow a > c (transitivity)
  3. If a<b∧c<d→a+c<b+da < b \land c < d \rightarrow a + c < b + d
  4. If a>b∧c>0→a⋅c>b⋅ca > b \land c > 0 \rightarrow a \cdot c > b \cdot c
  5. a<b→−a>−ba < b \rightarrow -a > -b
  6. If a>b∧c<0→a⋅c<b⋅ca > b \land c < 0 \rightarrow a \cdot c < b \cdot c
  7. ∀a∈R:a2≥0\forall a \in \mathbb{R}: a^2 \ge 0
  8. If 0≤a<b∧0≤c<d→a⋅c<b⋅d0 \le a < b \land 0 \le c < d \rightarrow a \cdot c < b \cdot d
  9. ∀a≠0:a and a−1\forall a \neq 0: a \text{ and } a^{-1} have the same sign
  10. ab>0↔((a>0∧b>0)∨(a<0∧b<0))ab > 0 \leftrightarrow \big( (a > 0 \land b > 0) \lor (a < 0 \land b < 0) \big)
  11. ab<0↔((a>0∧b<0)∨(a<0∧b>0))ab < 0 \leftrightarrow \big( (a > 0 \land b < 0) \lor (a < 0 \land b > 0) \big)
  12. Let ab>0ab > 0 (they have the same sign)

    →a<b⇔1b<1aa<x<b↔1b<1x<1a \rightarrow a < b \Leftrightarrow \frac{1}{b} < \frac{1}{a} \qquad a < x < b \leftrightarrow \frac{1}{b} < \frac{1}{x} < \frac{1}{a}

Additional Theorems

  1. Let a>b>0∧n∈Z+a > b > 0 \land n \in \mathbb{Z}^+: a>b→an>bna > b \rightarrow a^n > b^n
  2. Let a<b<0∧n∈Z+a < b < 0 \land n \in \mathbb{Z}^+:

    a<b→a2n>b2n a < b \rightarrow a^{2n} > b^{2n}

    a<b→a2n−1<b2n−1 a < b \rightarrow a^{2n-1} < b^{2n-1}

  3. Let a>0∧b>0a > 0 \land b > 0, then a<x<b→a2<x2<b2a < x < b \rightarrow a^2 < x^2 < b^2
  4. Let a<0∧b<0a < 0 \land b < 0, then a<x<b→a2>x2>b2a < x < b \rightarrow a^2 > x^2 > b^2
  5. Let a<0∧b>0a < 0 \land b > 0, then a<x<b→0≤x2<max⁡{a2;b2}a < x < b \rightarrow 0 \le x^2 < \max\{a^2; b^2\}

The Real Number Line

The real number line is a geometric representation that allows us to locate, compare, and order all real numbers.
Each point on the line corresponds to a unique real number, and conversely, each real number corresponds to a unique point on the line.

Intervals

An interval II (I⊂RI \subset \mathbb{R}) is a subset of real numbers that contains all the values between two given endpoints.

Open Interval

Does not include its endpoints.

a<x<b⇔x∈(a,b) a < x < b \quad \Leftrightarrow \quad x \in (a,b)

📝 Examples

(1,5)={x∈R∣1<x<5}(1,5)=\{x \in \mathbb{R} \mid 1<x<5\}

{% interval {"i":"(1,5)"} %}

(−3,2)={x∈R∣−3<x<2}(-3,2)=\{x \in \mathbb{R} \mid -3<x<2\}

{% interval {"i":"(-3,2)"} %}

(−12,4)={x∈R∣−12<x<4}\left(-\frac{1}{2},4\right)=\{x \in \mathbb{R} \mid -\frac{1}{2}<x<4\}

{% interval {"i":"(-0.5,4)"} %}

Closed Interval

Includes both endpoints.

a≤x≤b⇔x∈[a,b] a \le x \le b \quad \Leftrightarrow \quad x \in [a,b]

📝 Examples

[1,5]={x∈R∣1≤x≤5}[1,5]=\{x \in \mathbb{R} \mid 1\le x\le 5\}

{% interval {"i":"[1,5]"} %}

[−3,2]={x∈R∣−3≤x≤2}[-3,2]=\{x \in \mathbb{R} \mid -3\le x\le 2\}

{% interval {"i":"[-3,2]"} %}

[−12,4]={x∈R∣−12≤x≤4}\left[-\frac{1}{2},4\right]=\{x \in \mathbb{R} \mid -\frac{1}{2}\le x\le 4\}

{% interval {"i":"[-0.5,4]"} %}

Half-Open (Mixed) Intervals

Include only one of the endpoints.

a<x≤b⇔x∈(a,b] a < x \le b \quad \Leftrightarrow \quad x \in (a,b]

a≤x<b⇔x∈[a,b) a \le x < b \quad \Leftrightarrow \quad x \in [a,b)

📝 Examples

(1,5]={x∈R∣1<x≤5}(1,5]=\{x \in \mathbb{R} \mid 1<x\le5\}

{% interval {"i":"(1,5]"} %}

[1,5)={x∈R∣1≤x<5}[1,5)=\{x \in \mathbb{R} \mid 1\le x<5\}

{% interval {"i":"[1,5)"} %}

(−3,2]={x∈R∣−3<x≤2}(-3,2]=\{x \in \mathbb{R} \mid -3<x\le2\}

{% interval {"i":"(-3,2]"} %}

[−3,2)={x∈R∣−3≤x<2}[-3,2)=\{x \in \mathbb{R} \mid -3\le x<2\}

{% interval {"i":"[-3,2)"} %}

(−12,4]={x∈R∣−12<x≤4}\left(-\frac{1}{2},4\right]=\{x \in \mathbb{R} \mid -\frac{1}{2}<x\le4\}

{% interval {"i":"(-0.5,4]"} %}

[−12,4)={x∈R∣−12≤x<4}\left[-\frac{1}{2},4\right)=\{x \in \mathbb{R} \mid -\frac{1}{2}\le x<4\}

{% interval {"i":"[-0.5,4)"} %}

Infinite Intervals

When one of the endpoints is unbounded.

📝 Examples

(2,∞)={x∈R∣x>2}(2,\infty)=\{x \in \mathbb{R} \mid x>2\}

{% interval {"i":"(2,inf)"} %}

[2,∞)={x∈R∣x≥2}[2,\infty)=\{x \in \mathbb{R} \mid x\ge2\}

{% interval {"i":"[2,inf)"} %}

(−∞,3)={x∈R∣x<3}(-\infty,3)=\{x \in \mathbb{R} \mid x<3\}

{% interval {"i":"(-inf,3)"} %}

(−∞,3]={x∈R∣x≤3}(-\infty,3]=\{x \in \mathbb{R} \mid x\le3\}

{% interval {"i":"(-inf,3]"} %}

(−∞,−1)={x∈R∣x<−1}(-\infty,-1)=\{x \in \mathbb{R} \mid x<-1\}

{% interval {"i":"(-inf,-1)"} %}

[−4,∞)={x∈R∣x≥−4}[-4,\infty)=\{x \in \mathbb{R} \mid x\ge-4\}

{% interval {"i":"[-4,inf)"} %}

Bounded Below

x>a⇔x∈(a,∞) x > a \quad \Leftrightarrow \quad x \in (a,\infty)

x≥a⇔x∈[a,∞) x \ge a \quad \Leftrightarrow \quad x \in [a,\infty)

📝 Examples

(1,∞)={x∈R∣x>1}(1,\infty)=\{x \in \mathbb{R} \mid x>1\}

{% interval {"i":"(1,inf)"} %}

[0,∞)={x∈R∣x≥0}[0,\infty)=\{x \in \mathbb{R} \mid x\ge0\}

{% interval {"i":"[0,inf)"} %}

(−2,∞)={x∈R∣x>−2}(-2,\infty)=\{x \in \mathbb{R} \mid x>-2\}

{% interval {"i":"(-2,inf)"} %}

Bounded Above

x<b⇔x∈(−∞,b) x < b \quad \Leftrightarrow \quad x \in (-\infty,b)

x≤b⇔x∈(−∞,b] x \le b \quad \Leftrightarrow \quad x \in (-\infty,b]

📝 Examples

(−∞,4)={x∈R∣x<4}(-\infty,4)=\{x \in \mathbb{R} \mid x<4\}

{% interval {"i":"(-inf,4)"} %}

(−∞,0]={x∈R∣x≤0}(-\infty,0]=\{x \in \mathbb{R} \mid x\le0\}

{% interval {"i":"(-inf,0]"} %}

(−∞,−3)={x∈R∣x<−3}(-\infty,-3)=\{x \in \mathbb{R} \mid x<-3\}

{% interval {"i":"(-inf,-3)"} %}

Operations with Intervals

Let AA and BB be subsets of R\mathbb{R} (particularly, intervals).
The following operations are defined:

Union

The union contains all elements that belong to AA, to BB, or to both.

A∪B={x∈R∣x∈A ∨ x∈B} A \cup B = \{ x \in \mathbb{R} \mid x \in A \ \lor \ x \in B \}

Intersection

The intersection contains only the elements common to both sets.

A∩B={x∈R∣x∈A ∧ x∈B} A \cap B = \{ x \in \mathbb{R} \mid x \in A \ \land \ x \in B \}

Set Difference

The difference between AA and BB consists of the elements that belong to AA but not to BB.

A−B={x∈R∣x∈A ∧ x∉B} A - B = \{ x \in \mathbb{R} \mid x \in A \ \land \ x \notin B \}

Complement

The complement of AA consists of all real numbers that do not belong to AA.

AC=A′={x∈R∣x∉A} A^{C} = A' = \{ x \in \mathbb{R} \mid x \notin A \}

Inequality of Means

Let x1,x2,…,xn∈R+x_1, x_2, \dots, x_n \in \mathbb{R}^{+}. We define:

AM=x1+x2+⋯+xnn(Arithmetic Mean)AM = \frac{x_1 + x_2 + \cdots + x_n}{n} \qquad \text{(Arithmetic Mean)}

GM=x1⋅x2⋯xnn(Geometric Mean)GM = \sqrt[n]{x_1 \cdot x_2 \cdots x_n} \qquad \text{(Geometric Mean)}

HM=n1x1+1x2+⋯+1xn(Harmonic Mean)HM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}} \qquad \text{(Harmonic Mean)}

Then, the following fundamental inequality holds:

AM≥GM≥HM\boxed{AM \ge GM \ge HM}

Equivalently,

x1+x2+⋯+xnn≥x1⋅x2⋯xnn≥n1x1+1x2+⋯+1xn\boxed{\frac{x_1 + x_2 + \cdots + x_n}{n} \ge \sqrt[n]{x_1 \cdot x_2 \cdots x_n} \ge \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}}

Cauchy–Schwarz Inequality

Let a1,a2,…,ana_1, a_2, \dots, a_n and b1,b2,…,bnb_1, b_2, \dots, b_n be real numbers. Then:

(a1b1+a2b2+⋯+anbn)2≤(a12+a22+⋯+an2)(b12+b22+⋯+bn2)\boxed{\left(a_1 b_1 + a_2 b_2 + \cdots + a_n b_n\right)^2 \le \left(a_1^2 + a_2^2 + \cdots + a_n^2\right) \left(b_1^2 + b_2^2 + \cdots + b_n^2\right)}

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