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−1)(x−2)(x−3)≥0

Verified step by step guidance

1. Identify the critical points by setting each factor equal to zero: \(x - 1 = 0\), \(x - 2 = 0\), \(x - 3 = 0\). This gives the critical points \(x = 1\), \(x = 2\), and \(x = 3\).
2. Use these critical points to divide the real number line into intervals: \((-\infty, 1)\), \((1, 2)\), \((2, 3)\), and \((3, \infty)\).
3. Choose a test point from each interval and substitute it into the inequality \((x-1)(x-2)(x-3)\geq0\) to determine the sign of the product in each interval.
4. Analyze the sign of the product in each interval: If the product is positive, the interval is part of the solution set. If the product is negative, it is not.
5. Combine the intervals where the product is non-negative and include the critical points where the product equals zero to 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 must determine the intervals where the polynomial is either positive or negative. This often requires finding the roots of the polynomial and testing intervals between these roots.

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 intervals) or excluded (open intervals). For example, the interval [1, 3) includes 1 and goes up to, but does not include, 3.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the intervals where the polynomial inequality holds true. This involves marking the critical points (roots) and shading the regions that satisfy the inequality. Understanding how to accurately depict these intervals helps in interpreting the solution effectively.