# 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