Inequalities Formulas

Sir SagarJanuary 3, 2015January 3, 20150

Inequalities Formulas

1. Variables: x, y, z

Real numbers: $\left\{\begin{matrix} a, b, c, d& & \\ & &,m ,n \\ a_{1},a_{2},a_{3},....,a_{n}& & \end{matrix}\right.$

Determinants: D, $D_{x}, D_{y}, D_{z}$

2. Inequalities, Interval Notations and Graphs

3. If a > b, than b < a.

4. If a > b , than a – b > 0 or b – a < 0.

5. If a > b , than a + c > b +c.

6. If a > b , then a – c > b – c .

7. If a > b and c < d , then a + c > b + d.

8. If a > b and c > d, than a – d > b – c.

9. If a > b and m > 0, than ma > mb.

10. If a > b and m > 0 , than $\frac{a}{m} > \frac{b}{m}$ .

11. If 0 < a < b and n > 0 , than an < bn.

12. If 0 < a < b, than $\sqrt[n]{a}$ < $\sqrt[n]{b}$

13. $\sqrt{ab}$ ≤ $\frac{a+b}{2}$ ,

Where a > 0, b > 0; an equality is valid only if a = b.

14. a + $\frac{1}{a}$ ≥ 2, where a > 0; an equality takes place only at a = 1.

15. $\sqrt[n]{a_{1}a{2}……a-{n}}$ ≤ $\frac{a_{1}+a_{2}+….a_{n}}{n}$ ,where $a_{1},a_{2},…..a_{n} > 0.$

16. If ax + b > 0 and a > 0 , than x > $\frac{-b}{a}$ .

17. If ax + b > 0 and a < 0, than x < $\frac{-b}{a}$ .

18. $ax^{2}$ + bx +c > 0

19. $\left | a+b \right | \leq \left | a \right |+\left | b \right |$

20. If $\left | x \right |$ < a, than -a < x < a, where a > 0.

21. If $\left | x \right |$ > a, than x < -a and x > a where a > 0.

22. If x2 < a , than $\left | x \right |$ < $\sqrt{a}$ , where a > 0.

23. If x2 > a , than $\left | x \right |$ > $\sqrt{a}$ , where a > 0.

24. If $\frac{f(x)}{g(x)} > 0, than \left\{\begin{matrix} f(x)\cdot g(x)>0 & \\ g(x)\neq 0& \end{matrix}\right.$

25. If $\frac{f(x)}{g(x)} < 0, than \left\{\begin{matrix} f(x)\cdot g(x)< 0 & \\ g(x)\neq 0& \end{matrix}\right.$

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