Algebra - Inequalities

 Variables : x,y,z
Real number : { a,b,c,d,m,n     a1,a2,a3,...,an
Determinants : D,Dx,Dy,Dz

  1. Inequality Interval Notation Graph
    \(a \leq x \leq b\) \([a,b]\) \(-a-b \to x\)
    \(a <x \leq b\) \((a,b]\) \(-a-b \to x\)
    \(a \leq x < b\) \([a,b)\) \(-a-b \to x\)
    \(a < x < b\) \((a,b)\) \(-a-b \to x\)
    \(-\infty <x \leq b, x \leq b\) \((-\infty, b]\) \(-b \to x\)
    \(-\infty <x < b, x < b\) \((-\infty, b)\) \(-b \to x\)
    \(a\leq x< \infty, x \geq a\) \([ a, \infty)\) \(-a \to x\)
    \(a< x< \infty, x > a\) \((a, \infty)\) \(-a \to x\)
  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 \({a \over m}>{b \over m} .\)
  10. if a>b and m<0, then ma<mb.
  11. if a>b and m<0, then \({a \over m}<{b \over m} .\)
  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 \(\sqrt [n]{a} < \sqrt [n]{b}\)
  15. \(\sqrt {ab} \leq {a+b \over2},\)
    Where a>0 . b> 0; and equality is valid only if a= b .
  16. \(a+ {1 \over a} \geq 2,\)
    Where a>0; equality takes place only at a= 1 .
  17. \(\sqrt [n]{a_1a_2... a_n} \leq {a_1+a_2+...+a_n \over n}\), Where \(a_1,a_2,...,a_n > o.\)
  18. if ax+b > 0 and a > 0, theen x > \(-{b \over a}\).
  19. if ax+b > 0 and a < 0, theen x > \(-{b \over a}\).
  20. \(ax^2+bx + c > 0\)
  21. \(|a+b| \leq |a|+|b|\)
  22. if |X| < a , then -a < x < a , Where a > 0 .
  23. if |X| > a , Then a < - a and x > a , Where a > 0 . 
  24. if \(x^2 < a, \) then |x| < \(\sqrt a\), where a > 0 .
  25. if \(x^2 > a\), then |x| >\(\sqrt a\) , where a> 0
  26. if \({f(x) \over g(x) }> 0\), then  \(\{ f(x).g(x) > 0 , g(x) \neq 0\)
  27. if \({f(x) \over g(x) }< 0\), then  \(\{ f(x).g(x) < 0 , g(x) \neq 0\)