# Corollary: Rules of Calculations with Inequalities

(related to Definition: Order Relation of Real Numbers)

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

1. $x^2 > 0$ for all $$x\neq 0$$.
2. $$1 > 0$$.
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).
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).
5. Transitivity of the "$$<$$" (analogously the "$$>$$") relation
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).
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 "$$>$$").
8. If $$x > 0$$, then $$x^{-1} > 0$$.
9. If $$x < 0$$, then $$x^{-1} < 0$$.
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 $$y^{-1} < x^{-1}$$).
11. 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

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### References

#### Bibliography

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