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 x₁, x₂, x₃, ....x_n are
       solutions to the equation A = B (in ascending order from least to greatest), then

• x₁, x₂, x₃, ....x_n  are defined as the "Critical Numbers and
• Possible solutions to A<B are the intervals (x₁, x₂), (x₃, x₄), . . . (x_n-1, 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 x₁, x₂, x₃, ....x_n are
       solutions to the equation A = B (in ascending order from least to greatest), then

• x₁, x₂, x₃, ....x_n  are defined as the "Critical Numbers and
• Possible solutions to A<B are the closed intervals [x₁, x₂], [x₃, x₄], . . . [x_n-1, 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 x₁, x₂, x₃, ....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

• x₁, x₂, x₃, ....x_n  are defined as the "Critical Numbers and
• Possible solutions to A<B are the intervals (x₁, x₂), (x₃, x₄), . . . (x_n-1, 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 x₁, x₂, x₃, ....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

• x₁, x₂, x₃, ....x_n  are defined as the "Critical Numbers and
• Possible solutions to A<B are the intervals (x₁, x₂), (x₃, x₄), . . . (x_n-1, x_n) along with the endpoints of these intervals, provided the endpoint does not result in division by zero.