Summary: Compound and Absolute Value Inequalities

Key Concepts

  • Interval notation is a method to give the solution set of an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well.
  • Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality.
  • Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities.
  • Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value with the inequality symbol flipped.
    • [latex]|X|> k[/latex] which is equivalent to: [latex]X< -k\text{, or }X> k[/latex]
  • Absolute value inequality solutions can be verified by graphing. We can check the algebraic solutions by graphing as we cannot depend on a visual for a precise solution.


compound inequality
a problem or a statement that includes two inequalities
an interval describes a set of numbers where a solution falls
interval notation
a mathematical statement that describes a solution set and uses parentheses or brackets to indicate where an interval begins and ends
linear inequality
similar to a linear equation except that the solutions will include an interval of numbers