Let \(x,y,z,a,b\) be (if nothing else is stated) arbitrary real numbers. The following rules are valid for manipulating inequalities:
Rule 1)
For any \(x\neq 0\) it is \(x \cdot x >0\).
Rule 2)
\(1 > 0 \).
Rule 3)
From \(x > y\) it follows that \(x+a > y+a\) for any \(a\in\mathbb R\). (The rule is analogously valid and proven for any inequalities with “\( < \)” in between).
Rule 4)
If \(x > y\) and \(a > b\), then it follows that \(x+a > y+b\). (The rule is analogously valid and proven for any inequalities with “\( < \)” in between).
Rule 5)
Transitivity of the “\( < \)” (analogously the “\( > \)”) relation
Rule 6)
The inequality \(x > y\) does not change, if it is multiplied by any number \(a > 0\), i.e. it is \(ax > ay\). (The rule is analogously valid and proven for an inequalities with “\( < \)” in between).
Rule 7)
The inequality \(x > y\) changes, if it is multiplied by any number \(a < 0\), i.e. it is \(ax < ay\). (The rule is analogously valid and proven for an inequality with “\( < \)” in between, which then changes into “\( > \)”).
Rule 8)
If \(x > 0\), then \(x^{-1} > 0\).
Rule 9)
If \(x < 0\), then \(x^{-1} < 0\).
Rule 10)
If \(y > x > 0\), then \(x^{-1} > y^{-1} \). (The rule is analogously valid and proven for the inequality \(0 < x < y\), resulting in \(y^{-1} < x^{-1}\)).
Rule 11)
If \(0 \le x < y\) and \(0 \le a < b\) then \(ax < by\). (The rule is analogously valid and proven for \(by > ax\), following from \(y > x \ge 0\) and \(b > a \ge 0\)).
