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. First, identify the critical points by setting each factor equal to zero: solve \( x = 0 \), \( 4 - x = 0 \), and \( x - 6 = 0 \).
2. From these, find the critical points: \( x = 0 \), \( x = 4 \), and \( x = 6 \). These points divide the real number line into four intervals: \( (-\infty, 0) \), \( (0, 4) \), \( (4, 6) \), and \( (6, \infty) \).
3. Choose a test point from each interval and substitute it into the inequality \( x(4 - x)(x - 6) \leq 0 \) to determine whether the expression is positive or negative in that interval.
4. Include the critical points in the solution set if the inequality is \( \leq 0 \) (less than or equal to zero), because the expression equals zero at these points.
5. Combine the intervals where the inequality holds true and express the solution set in interval notation, then graph this solution on the real number line.

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 where the polynomial equals 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 and Graphing on the Number Line

Interval notation is a concise way to represent sets of numbers between two endpoints, using parentheses for exclusion and brackets for inclusion. Graphing the solution on a number line visually shows where the inequality is satisfied, marking critical points and shading the solution intervals.