Properties of Inequalities
Given that A, B, C and D represent real or complex algebraic
expressions, then the following are true:
Addition Properties of Inequality:
 If A < B, then A + C < B + C.
 If A > B, then A + C > B + C.
 If A ≤ B, then A + C
≤ B + C.
 If A ≥ B, then A + C ≥ B + C.
Subtraction Properties of Inequality:
 If A < B, then A  C < B  C.
 If A > B, then A  C > B  C.
 If A ≤ B, then A  C
≤ B  C.
 If A ≥ B, then A  C ≥ B  C.
Multiplication and Division By a Positive
Quantity:
 If A < B and C > 0, then AC < BC.
 If A > B, and C > 0, then AC > BC.
 If A < B and C > 0, then A/C < B/C.
 If A > B, and C > 0, then A/C > B/C.
 THESE PROPERTIES ALSO APPLY TO ≤ and ≥
Multiplication and Division By a Negative
Quantity: (Switch signs if multiplying or dividing by a negative)
 If A < B and C < 0, then AC > BC.
 If A > B, and C < 0, then AC <> BC.
 If A < B and C < 0, then A/C > B/C.
 If A > B, and C < 0, then A/C < B/C.
 THESE PROPERTIES ALSO APPLY TO ≤ and ≥
Critical Numbers Define Solution Intervals
If A < B or A > B is an inequality where A
and B are not rational expressions and x_{1}, x_{2},
x_{3}, ....x_{n} are
solutions to the equation A = B (in
ascending order from least to greatest), then
 x_{1}, x_{2}, x_{3}, ....x_{n}
are defined as the "Critical Numbers and
 Possible solutions to A<B are the intervals (x_{1}, x_{2}),
(x_{3}, x_{4}), . . . (x_{n1}, x_{n})
Critical Numbers Define Solution Intervals
If A ≤ B or A ≥ B is an inequality where A
and B are not rational expressions and x_{1}, x_{2},
x_{3}, ....x_{n} are
solutions to the equation A = B (in
ascending order from least to greatest), then
 x_{1}, x_{2}, x_{3}, ....x_{n}
are defined as the "Critical Numbers and
 Possible solutions to A<B are the closed intervals [x_{1},
x_{2}], [x_{3}, x_{4}], . . . [x_{n1},
x_{n}]
Critical Numbers Define Solution Intervals For
Rational Inequalities
If A/B < C/D or A/B > C/D is a rational
inequality and x_{1}, x_{2}, x_{3}, ....x_{n}
are
solutions to the equation A/B = C/D
or solutions to B=0 or D=0 (in ascending order from least to
greatest), then
 x_{1}, x_{2}, x_{3}, ....x_{n}
are defined as the "Critical Numbers and
 Possible solutions to A<B are the intervals (x_{1}, x_{2}),
(x_{3}, x_{4}), . . . (x_{n1}, x_{n})
Critical Numbers Define Solution Intervals For
Rational Inequalities
If A/B ≤ C/D or A/B ≥ C/D is a rational
inequality and x_{1}, x_{2}, x_{3}, ....x_{n}
are
solutions to the equation A/B = C/D
or solutions to B=0 or D=0 (in ascending order from least to
greatest), then
 x_{1}, x_{2}, x_{3}, ....x_{n}
are defined as the "Critical Numbers and
 Possible solutions to A<B are the intervals (x_{1}, x_{2}),
(x_{3}, x_{4}), . . . (x_{n1}, x_{n})
along with the endpoints of these intervals, provided the endpoint
does not result in division by zero.
