Mathematics - Chapter 1B: Inequalities

• Basic Rules:
• Inequalities are transitive (If a<b and b<c, then a<c)
• Addition and Subtraction does not change the inequality sign
• Multiplication and Division with negative number changes the inequality sign
• Taking reciprocal of two positive numbers always change the inequality sign

• Mistakes to avoid:
• Subtracting several inequalities
• Cross-multiplying inequalities
• Squaring both sides if the terms may have different sign

• Polynomial Inequalities:
• Factor the inequality, and draw graph to find the solution

• Rational Function Inequalities:
• Multiply LHS and RHS by the square of denominator, if it is not always >0.
• If denominator is always positive, it is possible to multiply both sides by the denominator.
• Note the restriction from the denominator (e.g. if the denominator is x-1 then x ≠1)

• Modulus Inequalities:
• Note the definition of mod(x).
• Some useful properties:
• | a | < b and b > 0 if and only if -b < a < b
• | a | > b > 0 if and only if a > b or a < -b
• | a | > | b | if and only if a­­­­2 > b2
• Easiest way is to graph the equations

• Substitution methods:
• Sometimes it is possible to replace a variable with another expression in a solution of an equation.
• Example: ‘Replace x by x+2 in the solution of x2 < 1+2x ’

• Notes:
• Round the value of x within the 5 s.f. range
• Only square both sides of the inequality if both sides are non-negative
• When doing a modulus inequality with analytical method, use ‘combining both cases, …’