Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. (x + 3)^3(2x - 1)(x + 4) ≥ 0

Verified step by step guidance

1. Step 1: First, we need to find the critical points of the inequality. The critical points are the values of x that make the expression equal to zero. So, set each factor of the polynomial to zero and solve for x. The factors are (x + 3), (2x - 1), and (x + 4).
2. Step 2: After finding the critical points, plot them on a number line. These points divide the number line into several intervals. We will test each interval to see if it makes the inequality true or false.
3. Step 3: Choose a test point in each interval and substitute it into the 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 'greater than or equal to', we include the critical points in our solution set. If the inequality was 'greater than', we would not include the critical points.
5. Step 5: Write the solution set in interval notation. The solution set is the union of all the intervals that make the inequality true.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to a value, typically zero, using inequality symbols such as ≥, ≤, >, or <. To solve these inequalities, one must determine the intervals where the polynomial is positive or negative, which often requires finding the roots of the polynomial and testing the sign of the polynomial in the resulting intervals.

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 intervals) or excluded (open intervals). For example, the interval [a, b] includes both a and b, while (a, b) does not. This notation is essential for expressing the solution set of inequalities succinctly.

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 identifying the roots and testing points in each interval, one can ascertain where the polynomial is greater than or equal to zero. This process is crucial for solving polynomial inequalities and helps in constructing the solution set in interval notation.