Question

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)\u003c0

Accepted Answer

First, identify the critical points by setting each factor equal to zero. For the inequality $$(1 - x)^2$ (x - \frac{5}{2}) \u003c 0$$, set $$(1 - x)^2$ = 0$$ and $$x - \frac{5}{2} = 0 to $$find the critical points. 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. 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 $$(1 - x)^2$ is always non-negative since it is squared. Test a sample value from each interval in the inequality to check whether the product is less than zero. For example, pick x=0$ for $$(-\infty, 1)$$, x=2$ for (1$$, \frac{5}{2})$$, and x=3$ for $$(\frac{5}{2}, \infty)$$. Based on the sign tests, determine which intervals satisfy the inequality (1$$ - $$x)^2 (x$$ - \frac{5}{2}) \u003c 0$$. Then express the solution set in interval notation and graph it on the real number line.

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

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Key Concepts

Polynomial Inequalities

Critical Points and Sign Analysis

Interval Notation and Graphing Solutions

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