Linear Inequality
A linear inequality in one variable can be written in the form
Ax + B < C,
where A, B, and C are real numbers, with A ≠ 0.
Addition Property of Inequality
For all real numbers A, B, and C, the inequalities
A < B and A + C < B + C
are equivalent.
In words, adding the same number to each side of an inequality does not change the solution set.
Multiplication Property of Inequality
For all real numbers A, B, and C, with C Z 0,
(a) the inequalities
A < B and AC < BC
are equivalent if C > 0;
(b) the inequalities
A < B and AC > BC
are equivalent if C < 0.
In words, each side of an inequality may be multiplied (or divided)
by a positive number without changing the direction of the inequality
symbol. Multiplying (or dividing) by a negative number requires that
we reverse the inequality symbol.