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Mathematics LibreTexts

2.3 Arithmetic of inequality

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 \(h \in  \mathbb{Z_+}\).

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  \(a< b\)  and   \(c< d\)  then   \(a+c< b+d\).