Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x(x+1)<12

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is compared to a value using inequality symbols (e.g., <, >, ≤, ≥). To solve these inequalities, one typically rearranges the inequality into standard form, finds the roots of the corresponding quadratic equation, and then tests intervals to determine where the inequality holds true.

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, (a, b) represents all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Testing Intervals

After finding the roots of a quadratic inequality, the number line is divided into intervals based on these roots. To determine where the inequality is satisfied, one selects test points from each interval and substitutes them back into the inequality. The results indicate whether the inequality holds true in that interval, allowing for the identification of the solution set.