How you show with calculus method inequality x^2/x +1<=-4 if x<=-2?

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crmhaske | College Teacher | (Level 3) Associate Educator

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First we will rearrange the function to bring all terms to the same side:



Using the calculus method, we must take the derivative of the function and show that f'(x) is decreasing for x<-2 and that x=-2 is a root of the function for f(x)<=0 for x<=-2:



Therefore, the function is decreasing for x<-2, and as such, x^2+4x+4<0 when x<-2.  As x=-2 is a root of the function, x^2+4x+4<=0 when x<=-2 is also true.