Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3+2x^2−x−2≥0

Polynomial inequalities involve expressions where a polynomial is compared to a value, typically zero, using inequality symbols such as ≥, ≤, >, or <. To solve these inequalities, one must find the values of the variable that make the inequality true, often by determining the roots of the polynomial and testing intervals between these roots.

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval [a, b) includes 'a' but not 'b', while (a, b) excludes both endpoints.

Graphing solution sets on a real number line visually represents the values that satisfy the inequality. This involves marking the roots of the polynomial and shading the appropriate regions based on the sign of the polynomial in each interval. This graphical representation helps in understanding the solution set and its relation to the inequality.