Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. 2x^3 - 7x^2 ≥ 3 - 8x

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions that compare a polynomial to a value using inequality symbols (e.g., ≥, ≤, >, <). To solve these inequalities, one typically finds the roots of the corresponding polynomial equation and tests intervals between these roots to determine where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses brackets and parentheses to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, [a, b] includes both a and b, while (a, b) does not include them.

Testing Intervals

After identifying the roots of a polynomial inequality, the next step is to test the sign of the polynomial in the intervals created by these roots. By selecting test points from each interval, one can determine whether the polynomial is positive or negative in those intervals, which helps in finding the solution set for the inequality.