Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. (x - 3)(x - 4)(x - 5)^2 ≤ 0

Verified step by step guidance

1. First, identify the critical points by setting each factor equal to zero: solve \(x - 3 = 0\), \(x - 4 = 0\), and \(x - 5 = 0\). These give the points \(x = 3\), \(x = 4\), and \(x = 5\).
2. Next, use these critical points to divide the real number line into intervals: \(( -\infty, 3 )\), \(( 3, 4 )\), \(( 4, 5 )\), and \(( 5, \infty )\).
3. Choose a test point from each interval and substitute it into the expression \((x - 3)(x - 4)(x - 5)^2\) to determine the sign (positive or negative) of the product in that interval.
4. Remember that \((x - 5)^2\) is always non-negative because it is squared, so it will not affect the sign change but will be zero at \(x = 5\). Use this to analyze the sign behavior around \(x = 5\).
5. Finally, combine the intervals where the product is less than or equal to zero, including points where the expression equals zero, and express the solution set in interval notation.

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 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 where the inequality holds.

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 set of inequalities.