# 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  $\{ f(x).g(x) < 0 , g(x) \neq 0$