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Chapter 2: Problem 33

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$(2-x)^{2}\left(x-\frac{7}{2}\right)<0$$

Short Answer

Expert verified
The solution to the inequality is \( ( 2 , \frac{7}{2} ) \).

Step by step solution

01

Simplifying the inequality

Let's simplify the given expression to basic polynomial form. The equation, \( (2-x)^{2}(x-\frac{7}{2})<0 \), can be rewritten as:\( (x-2)^{2}(x-\frac{7}{2})>0 \).
02

Finding the roots

Setting the inequality equal to zero will allow us to find the roots: \( (x-2)^{2}(x-\frac{7}{2})=0 \). The roots, from this equation, are \(x_1=2\) and \(x_2=\frac{7}{2}\).
03

Analyzing the number line

We'll plot on a number line the two roots we just found, splitting the real number line into three intervals: \(-\infty , 2\), \((2, \frac{7}{2})\), and \(\frac{7}{2} , \infty\) and then, we'll test a representative point from each of those intervals to determine whether the inequality is satisfied.
04

Checking the intervals

If we take test points: 0 from \(-\infty , 2\), 3 from \((2, \frac{7}{2})\), and 4 from \(\frac{7}{2} , \infty\), we get the following results:For \(-\infty , 2\) (test point: 0), the inequality becomes positive,For \(2, \frac{7}{2}\) (test point: 3), the inequality becomes negative, andFor \(\frac{7}{2} , \infty\) (test point: 4), again, the inequality becomes positive. Thus, our solution set is \(2 , \frac{7}{2}\).
05

Writing the answer in interval notation

Given the inequality is '<0', we exclude the roots and the final solution in interval notation would be :(2, \frac{7}{2}).

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