Corollary: Rules of Calculations with Inequalities

Let $x,y,z,a,b$ be (if nothing else is stated) arbitrary real numbers. The following rules are valid for manipulating inequalities:

$x^2 > 0$ for all $x\neq 0$.
$1 > 0 $.
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).
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 "$ < $" in between).
Transitivity of the "$ < $" (analogously the "$ > $") relation
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).
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 "$ < $" in between, which then changes into "$ > $").
If $x > 0$, then $x^{-1} > 0$.
If $x < 0$, then $x^{-1} < 0$.
If $y > x > 0$, then $x^{-1} > y^{-1} $. (The rule is analogously valid and proven for the inequality $0 < x < y$, resulting in $y^{-1} < x^{-1}$).
If $y > x > 0$, then $x^{-1} > y^{-1} $. (The rule is analogously valid and proven for the inequality $0 < x < y$, resulting in $y^{-1} < x^{-1}$).

Proofs: 1

Parts: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Propositions: 17 18 19
Sections: 20

Thank you to the contributors under CC BY-SA 4.0!

Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
Modler, F.; Kreh, M.: "Tutorium Analysis 1 und Lineare Algebra 1", Springer Spektrum, 2018, 4th Edition