Inequalities Formulas

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

notations

 

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

a

 

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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