Problem 33
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. $$ (2-x)^{2}\left(x-\frac{7}{2}\right)<0 $$
Step-by-Step Solution
Verified Answer
The solution to the inequality in interval notation is \((2,\frac{7}{2})\).
1Step 1: Factor the Inequality
Factor the given inequality \( (2-x)^{2}\left(x-\frac{7}{2}\right)<0 \). It's already factored so we proceed to the next step.
2Step 2: Find the Critical Points
The critical points are obtained by setting each factor equal to zero. Thus, for \(2-x = 0\), \(x = 2\) and for \(x-\frac{7}{2} = 0\), \(x = \frac{7}{2}\). Therefore, the critical points are \(2\) and \(\frac{7}{2}\).
3Step 3: Plot the Critical Points on a Number Line
Place the critical points on a number line. The number line will be divided into three intervals by the points \(2\) and \(\frac{7}{2}\).
4Step 4: Test the Intervals
Choose a number from each interval to evaluate the inequality. If the number makes the inequality true, then the interval is part of the solution set. For \(x<2\), we could choose \(1\); for \(2\frac{7}{2}\), we could choose \(4\). Using these test points produces \(9>0\), \(-4<0\), and \(9>0\) respectively.
5Step 5: Express the Solution in Interval Notation
The interval where the inequality holds true is when \(2
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