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. (1−x)^2(x−5/2)<0

Verified step by step guidance

1. First, identify the critical points by setting each factor of the inequality \((1-x)^2(x-\frac{5}{2}) < 0\) equal to zero. Solve \((1-x)^2 = 0\) and \(x-\frac{5}{2} = 0\).
2. Solve \((1-x)^2 = 0\) to find \(x = 1\). Since this is a squared term, it will be a repeated root.
3. Solve \(x-\frac{5}{2} = 0\) to find \(x = \frac{5}{2}\).
4. Use the critical points \(x = 1\) and \(x = \frac{5}{2}\) to divide the number line into intervals: \((-\infty, 1)\), \((1, \frac{5}{2})\), and \((\frac{5}{2}, \infty)\).
5. Test a point from each interval in the inequality \((1-x)^2(x-\frac{5}{2}) < 0\) to determine where the inequality holds true. Remember that \((1-x)^2\) is always non-negative, so focus on the sign of \(x-\frac{5}{2}\) in each interval.

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 typically finds the roots of the polynomial, which are the values that make the polynomial equal to zero, and then tests intervals between these roots to determine where the polynomial is positive or negative.

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 intervals) or excluded (open intervals). For example, the interval (1, 5] includes all numbers greater than 1 and up to and including 5.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the solutions to an inequality. Each interval where the inequality holds true is marked, often using open or closed circles to indicate whether endpoints are included. This graphical representation helps in understanding the range of values that satisfy the inequality.