Question 316238
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The corresponding polynomial has exactly three zeros: -7, -6, and 17.  That means the range of real values comprising the domain of the polynomial inequality is divided into four regions:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(-\infty,-7\right)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(-7,-6\right)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(-6,17\right)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(17,\infty\right)]


None of the endpoints are included because any of the finite endpoints would make the inequality equal zero.


Consider the sign of each factor for a selected value in each of the intervals:


From the first interval select -8:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (-)(-)(-)\ <\ 0]:  Invalid interval


From the second interval select -6.5:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (+)(-)(-)\ >\ 0]:  Values from this interval make the inequality true.


From the third interval select 0:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (+)(-)(+)\ <\ 0]:  Invalid interval


From the fourth interval select 18:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (+)(+)(+)\ >\ 0]:  Values from this interval make the inequality true.


The solution set is the interval:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(-7,-6\right)\ \large\ \cup\ \ \LARGE \left(17,\infty\right)]



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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