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(3−x)(x−5)≤0

Verified step by step guidance

1. First, identify the critical points by setting each factor equal to zero: solve \( x = 0 \), \( 3 - x = 0 \), and \( x - 5 = 0 \). These points divide the real number line into intervals.
2. The critical points are \( x = 0 \), \( x = 3 \), and \( x = 5 \). These points split the number line into four intervals: \( (-\infty, 0) \), \( (0, 3) \), \( (3, 5) \), and \( (5, \infty) \).
3. Choose a test point from each interval and substitute it into the expression \( x(3 - x)(x - 5) \) to determine the sign (positive or negative) of the product in that interval.
4. Since the inequality is \( x(3 - x)(x - 5) \leq 0 \), select the intervals where the product is less than or equal to zero. Remember to include the critical points where the expression equals zero.
5. Express the solution set as a union of intervals where the inequality holds true, and write the solution in interval notation. Finally, graph these intervals on the real number line, marking included endpoints with solid dots.

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. After solving the inequality, the solution set is expressed in interval notation and graphically shown on a number line to visualize the range of valid solutions.