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. (2−x)²(x−7/2)<0

Verified step by step guidance

1. First, identify the critical points by setting each factor equal to zero: solve \((2 - x)^2 = 0\) and \((x - \frac{7}{2}) = 0\) to find the values of \(x\) where the expression equals zero.
2. Next, determine the intervals on the real number line created by these critical points. These intervals will be where the inequality may change sign.
3. Choose a test point from each interval and substitute it into the inequality \((2 - x)^2 (x - \frac{7}{2}) < 0\) to check whether the expression is positive or negative in that interval.
4. Analyze the sign of each factor: note that \((2 - x)^2\) is always non-negative since it is squared, so the sign of the product depends mainly on \((x - \frac{7}{2})\).
5. Based on the sign analysis, write the solution set where the product is less than zero, and express this solution set in interval notation. Finally, sketch the solution on a real number line, marking the critical points and shading the appropriate intervals.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality signs (<, >, ≤, ≥). 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 the solution set as intervals on the real number line, using parentheses for open intervals and brackets for closed intervals. Graphing these intervals visually represents where the inequality is satisfied, aiding in understanding and communication of the solution.