Solve each polynomial inequality. Give the solution set in interval notation.
(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. Based on the inequality sign (≥ 0, > 0, ≤ 0, < 0), select the intervals where the expression satisfies the inequality. Remember to include or exclude the critical points depending on whether the inequality is strict or not.
5. Write the solution set in interval notation, combining the intervals where the inequality holds true, and include the critical points if the inequality is non-strict (≥ or ≤).

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 the solution set for the inequality.

Interval Notation

Interval notation is a concise way to represent sets of real numbers. It uses parentheses () for excluding endpoints and brackets [] for including them. This notation is essential for expressing the solution sets of inequalities clearly and precisely.