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Chapter 3: Problem 9

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. $$ x^{2}-6 x+9<0 $$

Short Answer

Expert verified
The solution set in interval notation is \((- \infty, 3) \cup (3, +\infty)\)

Step by step solution

01

Factor the Polynomial

In order to find the roots of the quadratic, begin by factoring the equation \(x^2 - 6x + 9\). Since it's a perfect square trinomial, factoring yields \((x-3)^2\).
02

Find Roots of the Polynomial

Setting \((x-3)^2\) equal to zero, and solving for x gives the root \(x=3\).
03

Test Intervals

Now, choose a number from each of the intervals \(-\infty, 3\) and \(3, +\infty\) and check it in the inequality \(x^2 - 6x + 9 < 0\). If the inequality is true, the entire interval is included in our solution. For \(x<3\), let's choose \(x=0\) as the test point. Substituting \(x=0\) in the inequality yields \(9 > 0\), which is true. For \(x>3\), choose \(x=4\) as the test point. Substituting \(x=4\) gives \(-7 < 0\), which is also true. However, since our original inequality is strict (i.e., it doesn't include equal to), the root \(x=3\) itself doesn't satisfy the inequality.
04

Write Solution in Interval Notation

As a result, you can write the solution set as \(-\infty, 3\) and \(3, +\infty\), or \((- \infty, 3) \cup (3, +\infty)\) in interval notation.
05

Graph the Solution

On a number line, show these ranges by shading everything to the left of \(3\) not including \(3\) itself and everything to right of \(3\) not including \(3\) itself.

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Most popular questions from this chapter

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=x^{3}(x+2)^{2}(x+1)$$

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Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the ZOOM OUT feature to show that f and g have identical end behavior. \(f(x)=x^{3}-6 x+1, g(x)=x^{3}\)
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