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

Verified step by step guidance

1. Identify the critical points by setting each factor equal to zero: solve \(x - 4 = 0\), \(2x + 3 = 0\), and \(3x - 1 = 0\). These points divide the number line into intervals.
2. Write down the critical points found: \(x = 4\), \(x = -\frac{3}{2}\), and \(x = \frac{1}{3}\). These will be the boundaries for testing intervals.
3. Determine the sign of the product \((x - 4)(2x + 3)(3x - 1)\) on each interval created by the critical points. Choose a test value from each interval and substitute it into the expression to check if the product is positive or negative.
4. Since the inequality is \(\geq 0\), include intervals where the product is positive or zero. Also, include the critical points themselves because the inequality allows equality.
5. Express the solution set in interval notation by combining all intervals where the product is nonnegative, including the critical points where the expression equals zero.

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 values of the variable that make each factor of the polynomial zero. These points divide the number line into intervals. By testing values from 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 numbers on the number line. It uses parentheses () for values not included and brackets [] for values included. After solving the inequality, the solution set is expressed in interval notation to clearly show all values satisfying the inequality.