Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. 23. (a) -x(x - 1)(x - 2) ≥ 0 (b) -x(x - 1)(x - 2) > 0 (c) -x(x - 1)(x - 2) ≤ 0 (d) -x(x - 1)(x - 2) < 0

Verified step by step guidance

1. Identify the critical points by setting the expression equal to zero: solve \(-x(x - 1)(x - 2) = 0\). The solutions are the values of \(x\) where the expression changes sign.
2. These critical points divide the real number line into intervals. List the intervals determined by the critical points: \((-\infty, 0)\), \((0, 1)\), \((1, 2)\), and \((2, \infty)\).
3. Choose a test point from each interval and substitute it into the expression \(-x(x - 1)(x - 2)\) to determine whether the expression is positive or negative on that interval.
4. For each inequality (a) \(\geq 0\), (b) \(> 0\), (c) \(\leq 0\), and (d) \(< 0\), use the sign information from the test points and include or exclude the critical points accordingly to write the solution set.
5. Express the solution sets in interval notation, remembering to use square brackets \([\ ]\) for inclusive inequalities (\(\geq\), \(\leq\)) and parentheses \((\ )\) for strict inequalities (\(>\), \(<\)).

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero 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 roots of the polynomial where the expression equals zero. These points divide the number line into intervals. By testing values in each interval, you determine whether the polynomial is positive or negative there, which helps identify solution sets for inequalities.

Interval Notation

Interval notation is a concise way to represent sets of real numbers between two endpoints. It uses parentheses for open intervals (excluding endpoints) and brackets for closed intervals (including endpoints). This notation clearly expresses the solution sets of inequalities.