- 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, …’
