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. Step 1: Set each factor of the inequality to zero and solve for x. This will give you the critical points. In this case, the factors are (x - 3), (x - 4), and (x - 5)^2. So, the critical points are x = 3, x = 4, and x = 5.
2. Step 2: Plot these critical points on a number line. These points divide the number line into intervals. In this case, the intervals are (-∞, 3), (3, 4), (4, 5), and (5, ∞).
3. Step 3: Choose a test point in each interval and substitute it into the original inequality. If the inequality is true, then all the numbers in that interval are solutions. If the inequality is false, then none of the numbers in that interval are solutions.
4. Step 4: Since the inequality is less than or equal to zero, the critical points are included in the solution set. So, the solution set is the union of all intervals that make the inequality true, including the critical points.
5. Step 5: Write the solution set in interval notation. Each interval is written as (a, b), [a, b], (a, b], or [a, b), where a and b are the endpoints of the interval. Parentheses are used when the endpoint is not included in the interval, and brackets are used when the endpoint is included. The union of intervals is represented by the symbol ∪.

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 values of the variable that make the polynomial less than or equal to zero. 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. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval [a, b) includes 'a' but not 'b', which is essential for expressing solution sets of inequalities clearly.

Sign Analysis

Sign analysis is a method used to determine the sign (positive or negative) of a polynomial across different intervals defined by its roots. By testing points in each interval, one can ascertain where the polynomial is less than or equal to zero, which is crucial for solving polynomial inequalities and identifying the correct solution set.