LessonMathematics · Order in IR

Order in IR

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Order relation in ℝ

Total order property

For all a, b ∈ $\mathbb{R}$ : a ≤ b or b ≤ a (at least one is true).

Transitivity

a ≤ b and b ≤ c ⟹ a ≤ c

a \le b et b \le c \Longrightarrow a \le c

Addition

a ≤ b ⟹ a + c ≤ b + c for all c ∈ $\mathbb{R}$ .

a \le b \Longrightarrow a + c \le b + c

Multiplication

a ≤ b and c > 0 ⟹ ac ≤ bc. If c < 0, the inequality flips: ac ≥ bc.

c < 0 : a \le b \Longrightarrow ac \ge bc

Upper and lower bounds

Upper bound

M is an upper bound of A if ∀x ∈ A, x ≤ M.

Supremum (sup)

The smallest upper bound. If A is bounded above, sup(A) exists in $\mathbb{R}$ .

Lower bound / Infimum

m is a lower bound if ∀x ∈ A, x ≥ m. inf(A) is the greatest lower bound.

Classic inequalities

Solving ax + b ≤ 0

If a > 0: x ≤ −b/a. If a < 0: x ≥ −b/a. Watch for direction change.

Square inequality

x² ≥ 0 for all x ∈ $\mathbb{R}$ . x² = 0 iff x = 0.

x^{2} \ge 0 \forall x \in \mathbb{R}

AM-GM inequality

For a, b ≥ 0: (a+b)/2 ≥ √(ab). Arithmetic mean ≥ geometric mean.

(a + b)/2 \ge √(ab) (a,b \ge 0)

Theorems - Order in ℝ

Theorem 1 - Order in $\mathbb{R}$

Application of theorem 1.

Theorem 2 - Order in $\mathbb{R}$

Application of theorem 2.

Theorem 3 - Order in $\mathbb{R}$

Application of theorem 3.

Theorem 4 - Order in $\mathbb{R}$

Application of theorem 4.

Theorem 5 - Order in $\mathbb{R}$

Application of theorem 5.

Review Sheet - Order in ℝ

Key Rule 1

Summary of point 1 of Order in $\mathbb{R}$ .

Key Rule 2

Summary of point 2 of Order in $\mathbb{R}$ .

Key Rule 3

Summary of point 3 of Order in $\mathbb{R}$ .

Key Rule 4

Summary of point 4 of Order in $\mathbb{R}$ .

Key Rule 5

Summary of point 5 of Order in $\mathbb{R}$ .

Key Rule 6

Summary of point 6 of Order in $\mathbb{R}$ .

Key Rule 7

Summary of point 7 of Order in $\mathbb{R}$ .

Essential references (Morocco Common Core Science)

Problem-solving method

Identify data, set formulas, solve step by step, then verify sign, unit, and coherence.

Common mistakes

Forgetting domain constraints, mixing equality/equivalence, and losing solutions during transformations.

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