Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x(x-1)≤6

Key Concepts

Quadratic Inequalities

Quadratic inequalities are expressions that involve a quadratic polynomial set in relation to a value, typically using symbols like ≤, ≥, <, or >. To solve these inequalities, one must first rearrange the expression to one side, resulting in a standard form that can be analyzed for its roots and intervals. The solution involves determining where the quadratic expression is less than or equal to the specified value.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval [a, b] includes both a and b, while (a, b) does not. This notation is essential for clearly expressing the solution set of inequalities.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of its linear factors. This process is crucial for solving quadratic equations and inequalities, as it allows for the identification of the roots or critical points where the expression changes sign. In the context of the given inequality, factoring helps to simplify the problem and determine the intervals where the inequality holds true.