Question 1120317
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Find all critical points:


Zeros:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{2}{x\ +\ 1}\ -\ \frac{4}{4x\ -\ 2}\ =\ 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 2]


Undefined Points:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{2}{x\ +\ 1}]


is undefined at *[tex \Large x\ =\ -1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{4}{4x\ -\ 2}]


is undefined at *[tex \Large x\ =\ \frac{1}{2}]


Divide the *[tex \Large x]-axis into 4 intervals:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -\infty\ <\ x\ <\ -1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -1\ <\ x\ <\ \frac{1}{2}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{2}\ <\ x\ <\ 2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2\ <\ x\ <\ \infty]


Select four values, one from each interval that is not an endpoint of the interval and define *[tex \Large f(x)\ =\ \frac{2}{x\ +\ 1}\ \ ]and *[tex \Large g(x)\ =\ \frac{4}{4x\ -\ 2}].


The original inequality is true wherever *[tex \Large f(x)\ -\ g(x)\ >\ 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(-2)\ -\ g(-2)\ =\ -1\ -\ \frac{2}{5}\ <\ 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(0)\ -\ g(0)\ =\ 2\ +\ 2\ >\ 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(1)\ -\ g(1)\ =\ 1\ -\ 2\ <\ 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(3)\ -\ g(3)\ =\ \frac{1}{2}\ -\ \frac{2}{5}\ >\ 0]


The solution set is then:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \{x\ \in\ \mathbb{R}\ |\ -1\ <\ x\ <\ \frac{1}{2}\}\ \text{U}\ \{x\ \in\ \mathbb{R}\ |\ x\ >\ 2\}] 
								
								
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<img src="http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg">
*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  
								
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