Identities and inequalities

Identities and inequalities

Definitions and Presentation

  • An identity is an equation that is true for all values of the variable.
  • An inequality shows the relative sizes or values of different expressions.
  • Inequalities can be strict (greater than or less than) or weak (greater than or equal to, or less than or equal to).
  • Identities are often signified by a three-bar equals symbol ≡, while inequalities use symbols like <, ≤, >, ≥.
  • Compound inequalities, such as a < x ≤ b, represent a range of values.

Expanding and Simplifying Identities

  • Identities can usually be simplified by expanding brackets, collecting like terms and factorising.
  • To expand brackets, particularly those with powers, it is useful to remember identities such as (a + b)² ≡ a² + 2ab + b² and (a - b)² ≡ a² - 2ab + b².
  • More complex expansions may involve the binomial theorem or the identities for the addition, subtraction, double, and half of angles in trigonometry.

Solving Inequalities

  • To solve a simple inequality, you can add, subtract, multiply, or divide both sides by the same amount, just like in equations. However, when multiplying or dividing by a negative number, the direction of the inequality must be reversed.
  • If an inequality contains x terms on both sides, it might be useful to collect all the terms on one side, leaving 0 on the other.
  • Quadratic inequalities can be solved by finding the roots of the equivalent equation, then testing each interval determined by these roots.
  • Inequalities with absolute values can be split into two separate inequalities to solve.
  • Once solved, inequalities can be depicted on a number line.
  • To combine multiple inequalities, find the common values that satisfy all the inequalities.

Applications of Identities and Inequalities

  • Identities are used in algebra to simplify and factor expressions, to solve equations, and to rewrite expressions in alternate forms.
  • Inequalities are used in practical situations where there is a range of possible solutions, such as in optimisation problems and probability.
  • They also help in approximation, rounding, and error analysis, as these usually involve a range of values rather than a single value.