Inequalities

Posted In: Mathematics

variables: x, y, z
Real numbers: a, b, c, d …., m, n, … a1, 2, 3, … n
Determinants: D, Dx, y, z

  1. Inequalities vs Interval notations:
    • a ≤ x ≤ b      [a,b]
    • a < x ≤ b      (a,b]
    • a ≤ x < b      [a,b)
    • a < x < b      (a,b)
    • -∞ < x ≤ b, x ≤ b     (-∞,b]
    • a ≤ x < ∞, a ≤ x     [a,∞)
    • a < x < ∞, a < x     (a,∞)
  2. if a > b, then b < a
  3. if a > b, then a-b > 0 or b-a < 0
  4. if a > b, then a+c > b+c
  5. if a > b, then a-c > b-c
  6. if a > b and c > d, then a+c > b+d
  7. if a > b and c > d, then a-d > b-c
  8. if a > b and m > 0, then ma > mb
  9. if a > b and m > 0, then $a/m>b/m$
  10. if a > b and m < 0, then ma < mb
  11. if a > b and m < 0, then $a/m<b/m$
  12. if 0 < a < b and n > 0, then $a^n<b^n$
  13. if 0 < a < b and n < 0, then $a^n>b^n$
  14. if 0 < a < b, then $√^na<√^nb$
  15. $√{ab}≤{a+b}/2$,
    where a > 0, b > 0;
    an equality is valid only if a = b
  16. $a+1/a≥2$,
    where a > 0; an equality takes place only at a=1
  17. $√{a_1a_2…a_n}≤{a_1+a_2+…+a_n}/n, \where a_1,a_2,…,a_n>0$
  18. if $ax+b>0$ and $a>0$, then $x>-b/a$
  19. if $ax+b>0$ and $a<0$, then $x<-b/a$
  20. $|a+b|≤|a|+|b|$
  21. if $|x|< a$, then $-a< x< a$, where $a>0$
  22. if $|x|>a$, then $x<-a$ and $x>a$, where $a>0$
  23. if $x^2< a$, then $|x|<√a$, where $a>0$
  24. if $x^2>a$, then $|x|>√a$, where $a>0$
  25. if ${f(x)}/{g(x)}>0$, then $\{\table f(x)g(x)>0;g(x)≠0$
  26. if ${f(x)}/{g(x)}<0$, then $\{\table f(x)g(x)<0;g(x)≠0$

Post Tags: Algebra


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