1. Combination of inequalities: If a < b and b < c, then a < c.
2. Adding a constant: We can add or subtract a constant to both sides of an inequality and preserve the direction of the inequality: If a < b and c is any number, then
   
   a + c < b + c, and
   
   a - c < b - c
3. Addition of inequalities: If we add inequalities in the same direction, the result is an inequality in the same direction:
   
   If a < b and c < d, then a + c < b + d.
4. Multiplication by a positive number: If we multiply both sides of an inequality by a positive number, the result is an inequality in the same direction:
   
   If a < b and c is positive, then ac < bc.
5. Multiplication by a negative number: If we multiply both sides of in inequality by a negative number, the result is an inequality in the opposite direction:
   
   If a < b and c is negative, then ac > bc.