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. 5x≤2−3x^2

Polynomial inequalities involve expressions where a polynomial is compared 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.

Graphing solution sets on a real number line visually represents the values that satisfy the inequality. This involves marking the critical points (roots) and shading the appropriate regions based on whether the inequality is strict or inclusive, helping to clearly communicate the solution set.

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 interval) or excluded (open interval). For example, the interval [a, b) includes 'a' but not 'b', which is essential for expressing the solution set of inequalities accurately.