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−x)(x−5)≤0

Verified step by step guidance

1. Identify the critical points by setting each factor of the polynomial equal to zero: \(x = 0\), \(3-x = 0\), and \(x-5 = 0\).
2. Solve each equation to find the critical points: \(x = 0\), \(x = 3\), and \(x = 5\).
3. Use the critical points to divide the number line into intervals: \((-\infty, 0)\), \((0, 3)\), \((3, 5)\), and \((5, \infty)\).
4. Choose a test point from each interval and substitute it into the inequality \(x(3-x)(x-5)\leq0\) to determine if the interval satisfies the inequality.
5. Combine the intervals that satisfy the inequality and express the solution set in interval notation.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero using inequality signs (e.g., ≤, ≥, <, >). To solve these inequalities, one typically finds the roots of the polynomial, which are the values that make the polynomial equal to zero, and then determines the intervals where the polynomial is positive or negative.

Interval Notation

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 interval) or excluded (open interval). For example, (a, b) means all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the solutions to an inequality. Each interval where the inequality holds true is marked, often using solid or open circles to indicate whether endpoints are included or excluded. This graphical representation helps in understanding the range of values that satisfy the inequality.