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 equal to zero. For the inequality \((1 - x)^2 (x - \frac{5}{2}) < 0\), set \((1 - x)^2 = 0\) and \(x - \frac{5}{2} = 0\) to find the critical points.
2. Solve each equation: \((1 - x)^2 = 0\) gives \(x = 1\), and \(x - \frac{5}{2} = 0\) gives \(x = \frac{5}{2}\). These points divide the real number line into intervals to test.
3. Determine the sign of the expression \((1 - x)^2 (x - \frac{5}{2})\) on each interval created by the critical points: \((-\infty, 1)\), \((1, \frac{5}{2})\), and \((\frac{5}{2}, \infty)\). Remember that \((1 - x)^2\) is always non-negative since it is squared.
4. Test a sample value from each interval in the inequality to check whether the product is less than zero. For example, pick \(x=0\) for \((-\infty, 1)\), \(x=2\) for \((1, \frac{5}{2})\), and \(x=3\) for \((\frac{5}{2}, \infty)\).
5. Based on the sign tests, determine which intervals satisfy the inequality \((1 - x)^2 (x - \frac{5}{2}) < 0\). Then express the solution set in interval notation and graph it on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero using inequality signs like <, >, ≤, or ≥. 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 values where the polynomial equals zero or is undefined, dividing the number line into intervals. By testing points in each interval, you determine whether the polynomial is positive or negative there, which helps identify where the inequality holds.

Interval Notation and Graphing Solutions

Interval notation expresses solution sets as ranges of values, using parentheses for strict inequalities and brackets for inclusive ones. Graphing on a number line visually represents these intervals, showing where the polynomial inequality is satisfied.