Equality - two equations or quantities are of equal and the same value.
Inequality - two equations or quantities are lesser or greater than each other.
Basic Laws of Equality
- Reflexive Property: x = x
- Symmetric Property: if x = y, then y = x
- Transitive Property:
- if x = y, and y = z, then, x = z. (x = y = z)
- if w = x, and y = z, then, w + y = x + z.
- if w = x, and y = z, then, wy = xz.
- First Conclusion: things equal to the same thing are equal to each other.
- Second Conclusion: equals added to equals are equals
- Third Conclusion: equals multiplied to equals are equals
Basic Laws of Inequality
- x > y; x is greater than y
- x < y; x is less than y
- x </= y; x is less than or equal to y
- x>/=y; x is greater than or equal to y
Basic Theorems on Inequalities
- x > y if and only if -x < -y
- if x > 0, then -x < 0
- if -x > 0, then x < 0
- if x > y, z < 0, then ac < bc
- if x > y, w > z, then (x + w) > (y + z)
- if x > y, w> z, and w, x, y, z > 0, then xw > yz
- if x > 0, y > 0, x > y, then 1/x < 1/y