2.3: Arithmetic of inequality
- Page ID
- 7427
This page is a draft and is under active development.
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Definition: Inequality
Let \(a, b\in \mathbb{Z}\). Then
- \(a< b\) provided \(b=a + k\), for some \(k \in \mathbb{Z_+}\).
- \(a> b\) provided \(a=b + h\), for some \(h \in \mathbb{Z_+}\).
Theorem \(\PageIndex{1}\)
Let \(a, b\in \mathbb{Z}\).
- If \(a< b\) then \(a+c< b+c\), \( \forall c \in \mathbb{Z}\).
- If \(a< b\) then \(ac< bc\),\( \forall c \in \mathbb{Z_+}\).
- If \(a< b\) then \(ac> bc\),\( \forall c \in \mathbb{Z_-}\).
- If \(a< b\) and \(c< d\) then \(a+c< b+d\).
- Proof
-
Let \(a, b, c \in \mathbb{Z}\) such that \(a <b\). Then \(b=a + k\), for some \(k \in \mathbb{Z_+}\).
1. Now consider, \(b+c= (a+k)+c= (a+c)+k\), for some \(k \in \mathbb{Z_+}\). Thus \(a+c< b+c\).
2.
Example \(\PageIndex{1}\):
Determine all integers \(m\) that satisfy \(-12m \geq 324\).
Solution
Since \(-12m \geq 324\), \( m \leq -\dfrac{324}{12}=-27\).
Example \(\PageIndex{2}\):
Determine all integers \(m\) that satisfy \(14m \geq 635\).
Solution
Since \(14m \geq 635\), \( m \geq \dfrac{635}{14}=45.35\). Thus the solutions are \( \{m\in \mathbb{Z}| m\geq 46\}.\)
Example \(\PageIndex{3}\):
Determine all integers \(k\) that satisfy \( -165+ 98k \geq 0, -335+199k \geq 0, -165+ 98k < 100 \) and \( -335+199k <100\).
Solution
Since \(-165 + 98k ≥ 0, k ≥ 1.68\).
Since \(-335 + 199k ≥ 0, k ≥ 1.68\).
Since \(-165 + 98k < 100, 98k < 265, \) and \( k < 2.70.\)
Since \(-335 + 199k < 100, 199k < 435,\) and \( k < 2.18.\)
Since \( 1.68 ≤ k < 2.18\) and \( k ∈ ℤ, k = 2. \)