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

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1. Identify the critical points by setting each factor equal to zero: solve \(x - 1 = 0\), \(x - 2 = 0\), and \(x - 3 = 0\). These points divide the real number line into intervals.
2. The critical points are \(x = 1\), \(x = 2\), and \(x = 3\). These points split the number line into four intervals: \(( -\infty, 1 )\), \((1, 2)\), \((2, 3)\), and \((3, \infty)\).
3. Choose a test point from each interval and substitute it into the expression \((x - 1)(x - 2)(x - 3)\) to determine the sign (positive or negative) of the product in that interval.
4. Since the inequality is \(\geq 0\), include intervals where the product is positive or zero. Also, include the critical points themselves because the inequality allows equality.
5. Express the solution set by combining the intervals where the product is non-negative, and write the final answer in interval notation. Then, graph this solution set on the real number line, marking included points with solid dots.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols (>, <, ≥, ≤). Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Critical Points and Sign Analysis

Critical points are the values of the variable where the polynomial equals zero, dividing the number line into intervals. By testing points in each interval, you determine whether the polynomial is positive or negative there, which helps identify where the inequality holds.

Interval Notation and Graphing on the Number Line

Interval notation expresses the solution set using intervals that show where the inequality is true, using parentheses or brackets to indicate whether endpoints are included. Graphing on the number line visually represents these intervals, aiding in understanding the solution's range.