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Mathematics LibreTexts

2.3: Arithmetic of inequality

  • Page ID
    7427
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    Definition

    Let \(a, b\in \mathbb{Z}\). Then

    1. \(a< b\) provided \(b=a + k\), for some \(k \in \mathbb{Z_+}\).
    2. \(a> b\) provided \(a=b + h\), for some \(h \in \mathbb{Z_+}\).

    Theorem \(\PageIndex{1}\)

    Let \(a, b\in \mathbb{Z}\).

    1. If \(a< b\) then \(a+c< b+c\), \( \forall c \in \mathbb{Z}\).
    2. If \(a< b\) then \(ac< bc\),\( \forall c \in \mathbb{Z_+}\).
    3. If \(a< b\) then \(ac> bc\),\( \forall c \in \mathbb{Z_-}\).
    4. 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 \( k ∈ ℤ, k = 2.  \)