Algebraic Inequalities (AS)

Understanding algebraic inequalities is crucial for solving mathematical problems within the further statistics syllabus. An inequality shows the relationship between two expressions that may not be equal. The symbols used include greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤).
When handling algebraic inequalities, it’s essential to remember that reversing the inequality sign is necessary when multiplying or dividing by a negative number. For instance, if -x > 5, then x < -5.
It’s also crucial to be aware of the distinction between open and closed inequalities. An open inequality does not include the endpoints (represented as < or >), while a closed inequality does include the endpoints (represented as ≤ or ≥).
A vital concept in understanding inequalities is interval notation. The interval of values that satisfy an inequality can be represented on a number line or in interval notation. For instance, x ≤ 5 is represented as (-∞, 5] in interval notation.
For multiple inequalities, the conjunctions ‘and’ and ‘or’ are used. ‘And’ refers to the overlap or intersection of solutions, while ‘or’ includes all solutions from each inequality. Solve each inequality separately when working with ‘or’ statement, and find the intersection when dealing with ‘and’.
Quadratic inequalities can be solved graphically by drawing the parabola and finding the regions where it is above or below the x-axis, depending on the direction of the inequality.
For inequalities involving absolute values, remember that the expression inside the absolute value is reflecting distance from zero, hence can be positive or negative. Therefore, break down the inequality into two separate cases to solve.
When manipulating rational inequalities, find the critical values (the zeros of the numerator and denominator), form intervals based on these critical values, test a number from each interval, and then combine intervals that satisfy the inequality.

Keep practicing different problems involving algebraic inequalities to get comfortable with their rules and nuances. Strong algebraic skills will be beneficial for handling statistical concepts that are yet to come.