Properties of the Numerical Inequalities

For any real numbers `a, b, c, d` is used the following properties:

  1. If `a>b`, then `b<a`.
  2. If ` ` ` ``a>b` and `b>c`, then `a>c` (the transitive property).
  3. If `a>b` then `a+c>b+c`.
  4. If `a>b` and `c` - positive number (`c>0`), then `ac>bc` (if the both parts of correct inequality to multiply by the same positive number, then we obtain correct inequality)
  5. If `a>b` and c - negative number (`c<0`), then `ac<bc` (if the both parts of correct inequality to multiply by the same negative number and change the sign of the initial inequality in the opposite, then we obtain correct inequality).
  6. If `a>b` and `c>d`, then `a+c>b+d` ( if two correct inequalities add termwise, we will obtain correct inequality).
  7. If `a, b, c, d ` - positive numbers and `a>b` and `c>d`, then `ac>bd` ( if the correct inequality of same sign to multiply termwise, the left and right parts of which are positive numbers, we will obtain correct inequality).
  8. If `a>b` and `c<d`, then `a-c>b-d`.
  9. If `a>b>0`, then `1/a<1/b` .
  10. If `a>b>0`, then `a^n>b^n` for any naturall `n`.