### Inequalities

### Definition

ℝ

ℝ

{0} = The single point 0

^{+}= {a | a ∈ ℝ and a is located to the R.H.S. of 0 in the number line}ℝ

^{-}= {a | a ∈ ℝ and a is located to the L.H.S. of 0 in the number line}{0} = The single point 0

Then, ℝ = ℝ

^{-}∪ {0} ∪ ℝ^{+}### Definition

Let a, b ∈ ℝ

- a - b ∈ ℝ
^{+}, We say that a is greater than b. When this happens, we write: a > b - a - b ∈ ℝ
^{+}∪ {0}, We say that a is greater than or equal to b. When this happens, we write: a ≥ b - a - b ∈ ℝ
^{-}, We say that a is less than b. When this happens, we write: a < b - a - b ∈ ℝ
^{-}∪ {0}, We say that a is less than or equal to b. When this happens, we write: a ≤ b

r ∈ ℝ

r ∈ ℝ

r ∈ ℝ

r ∈ ℝ

^{+}⇒ r - 0 ∈ ℝ^{+}⇒ r > 0r ∈ ℝ

^{+}∪ {0} ⇔ r ≥ 0r ∈ ℝ

^{-}⇔ r < 0r ∈ ℝ

^{-}∪ {0} ⇔ r ≤ 0### Convection

a ∈ ℝ

a ∈ ℝ

a ∈ ℝ

a ∈ ℝ

^{+}& b ∈ ℝ^{+}⇒ ab ∈ ℝ^{+}a ∈ ℝ

^{+}& b ∈ ℝ^{-}⇒ ab ∈ ℝ^{-}a ∈ ℝ

^{-}& b ∈ ℝ^{+}⇒ ab ∈ ℝ^{-}a ∈ ℝ

^{-}& b ∈ ℝ^{-}⇒ ab ∈ ℝ^{+}### Theorem

Let a, b, c ∈ ℝ

- a > b ⇒ ra > rb ∀r > 0 & ra < rb ∀r < 0
- a ≥ b ⇒ ra ≥ rb ∀r > 0 & ra ≤ rb ∀r < 0
- a < b ⇒ ra < rb ∀r > 0 & ra > rb ∀r < 0
- a ≤ b ⇒ ra ≤ rb ∀r > 0 & ra ≥ rb ∀r < 0
- a > b & b > c ⇒ a > c
- a < b & b < c ⇒ a < c
- a > b ⇒ a + d > b + d & a - d > b - d ∀d ∈ ℝ
- a ≥ b ⇒ a + d ≥ b + d & a - d ≥ b - d ∀d ∈ ℝ
- a < b ⇒ a + d < b + d & a - d < b - d ∀d ∈ ℝ
- a ≤ b ⇒ a + d ≤ b + d & a - d ≤ b - d ∀d ∈ ℝ

### Proof:

Let a, b ∈ ℝ

a > b ⇒ ra > rb ∀r > 0 & ra < rb ∀r < 0

Let a > b & r > 0

Then a - b ∈ ℝ

r (a - b) ∈ ℝ

ra - rb ∈ ℝ

ra > rb [by Definition]

Now let a > b & r < 0

Then a - b ∈ ℝ

r (a - b) ∈ ℝ

ra - rb ∈ ℝ

ra < rb [by Definition]

a > b ⇒ ra > rb ∀r > 0 & ra < rb ∀r < 0

Let a > b & r > 0

Then a - b ∈ ℝ

^{+}& r ∈ ℝ^{+}r (a - b) ∈ ℝ

^{+}[Convection]ra - rb ∈ ℝ

^{+}[Distributive Law]ra > rb [by Definition]

Now let a > b & r < 0

Then a - b ∈ ℝ

^{+}& r ∈ ℝ^{-}r (a - b) ∈ ℝ

^{-}[Convection]ra - rb ∈ ℝ

^{-}[Distributive Law]ra < rb [by Definition]

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