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

Verified step by step guidance

1. First, 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. Next, determine the intervals created by these critical points. The intervals will be \(( -\infty, \text{first root} )\), \(( \text{first root}, \text{second root} )\), \(( \text{second root}, \text{third root} )\), and \(( \text{third root}, +\infty )\).
3. Choose a test point from each interval and substitute it into the expression \((x - 4)(2x + 3)(3x - 1)\) to check whether the product is positive or negative in that interval.
4. Based on the sign of the product in each interval and the inequality \(\geq 0\), determine which intervals satisfy the inequality. Remember to include points where the product equals zero because of the \(\geq\) sign.
5. Finally, write the solution set in interval notation, combining all intervals where the inequality holds true, 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 signs (>, <, ≥, ≤). 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 a value 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 real numbers. 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.