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

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.