Inequalities

• Inequalities are mathematical expressions involving the symbols < (less than), ≤ (less than or equal to), > (greater than), or ≥ (greater than or equal to), and ≠ (not equal to).

• In the context of an inequality, there are potentially multiple valid solutions, not just a single answer.

• Inequalities can be solved similarly to equations, yet, whenever you multiply or divide both sides of an inequality by a negative number, the inequality symbol must be flipped.

• If multiple inequalities are linked by the word ‘and’, then only the values that satisfy both inequalities are valid. This is referred to as the intersection of two sets.

• When inequalities are combined by the word ‘or’, the solution will be all the values that satisfy either of the inequalities. This is called the union of two sets.

• For a number line representation, exclusive inequalities (<, >) are denoted with an open circle, while inclusive inequalities (≤, ≥) with a closed circle.

• When dealing with quadratic inequalities, you first need to make the inequality equal to zero and then solve the equation.

• Unfamiliar and complex inequalities may sometimes be made simpler by substitifying (x = y^2).

• Be careful about strict inequalities when square rooting both sides of an inequality.

• Inequalities play a vital role in real-world scenarios such as business modelling, scientific research, and engineering.

• Collectively, algebraic skills involving inequalities are fundamental to making reasoned decisions about ranges of possible values in diverse contexts.

• Practice is essential to gaining proficiency in solving and applying inequalities. Solving a variety of inequality equations will help enhance logical thinking skills.

• Always check your answers by substituting them back into the original inequality. If both sides of the inequality are true, then the solution is correct.

• Never forget to write the solution in the format required in the instructions (e.g., in interval notation or as an inequality).