Algebra: Laws and Theorems of Equality and Inequality

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