Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x(4−x)(x−6)≤0

Verified step by step guidance

1. Identify the critical points by setting each factor of the polynomial equal to zero: \(x = 0\), \(4 - x = 0\), and \(x - 6 = 0\).
2. Solve for \(x\) in each equation to find the critical points: \(x = 0\), \(x = 4\), and \(x = 6\).
3. Use the critical points to divide the number line into intervals: \((-\infty, 0)\), \((0, 4)\), \((4, 6)\), and \((6, \infty)\).
4. Choose a test point from each interval and substitute it into the inequality \(x(4-x)(x-6) \leq 0\) to determine the sign of the expression in each interval.
5. Determine which intervals satisfy the inequality and express the solution set in interval notation, including endpoints where the inequality is \(\leq\).

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 to see where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval [a, b] includes both a and b, while (a, b) does not include them. This notation is essential for expressing the solution set of inequalities succinctly.

Graphing on a Number Line

Graphing solutions on a number line visually represents the set of values that satisfy the inequality. Each solution is marked with a solid dot for included endpoints and an open dot for excluded endpoints. This graphical representation helps in understanding the solution set's extent and is a useful tool for visual learners to grasp the concept of inequalities and their solutions.