There are a number of properties that are important for understanding how to evaluate mathematical expressions, equations and inequalities.
Expressions, like [4x], do not involve the use of equivalence [=], while equations, like [4x = 4] equate two expressions and inequalities represent differences on either side of the inequalitiy, as in [4x > 2].
Properties of Expressions:
Distributive Property - For any real numbers, [a], [b], and [c], [a(b + c) = ab + ac] and [(b + c)a = ba + ca] and [ab + ac = a(b + c)].
Inverse Properties - For any real number, [a], there is a single real number [-a], such that [a + (-a) = 0] and [-a + a = 0].
For any nonzero real number [a], there is a single real number [1/a] such that [a x 1/a = 1] and [1/a x a = 1].
Identity Properties - For any real number, [a], [a + 0 = 0 + a = a] and [a x 1 = 1 x a = a].
Commutative Properties - For any real numbers, [a], [b], [a + b = b + a] and [ab = ba].
Associative Properties - For any real numbers, [a], [b], and [c], [a + (b + c) = (a + b) + c] and [a(bc) = (ab)c].
Multiplication Property of Zero - For all real numbers [a], [a x 0 = 0] and [0 x a = 0].
Properties of Equations:
Addition Property of Equality - For all real numbers [a], [b], and [c], the equations [a = b] and [a + c = b + c] are equivalent.
Multiplication Property of Equality - For all real numbers [a], [b], and [c], where [c 
0], [a = b] and [ac = bc] are equivalent.
Properties of Inequalities:
Addition Property of Inequality - For all real numbers [a], [b], and [c], the inequalities [a < b] and [a + c < b + c] are equivalent.
This is true for the greater than [>] sign, the less than or equal sign [
] and the greater than or equal sign [
].
Multiplication Property of Inequality - For all real numbers [a], [b], and [c], where [c 0], the inequalities [a < b] and [ac > bc] are equivalent if [c] is greater than zero.
This is true for the greater than [>] sign, the less than or equal sign [] and the greater than or equal sign [].
