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Find the domain of:

$$\frac{1}{\sqrt{3-|5-\frac{2}{x}|}}$$

I really don't know how to start.

Can anyone help?

Thanks.

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  • $\begingroup$ Your function is not at all clear. You should either typeset in MathJax (best) or type a formula with many parentheses and let us do the typesetting. Otherwise I do not know which function you mean. $\endgroup$ – Rory Daulton Feb 7 '15 at 11:38
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    $\begingroup$ Did you mean $\dfrac{1}{\sqrt{3-\left| 5-\dfrac{2}{x} \right|}}$? $\endgroup$ – user164524 Feb 7 '15 at 11:38
  • $\begingroup$ @Mathematician171 yes. $\endgroup$ – Hong Yuan Feb 7 '15 at 11:48
  • $\begingroup$ @RoryDaulton sorry, I kinda new here :3 second comment shows. $\endgroup$ – Hong Yuan Feb 7 '15 at 11:49
  • $\begingroup$ meta.math.stackexchange.com/questions/5020/… $\endgroup$ – Timbuc Feb 7 '15 at 11:56
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You must solve the inequality

$$3-\left|5-\frac2x\right|>0\iff\left|5-\frac2x\right|<3\iff -3<5-\frac2x<3$$

Now:

$$-3<5-\frac2x\implies 8\;\frac{x-\frac14}x>0\iff \color{red}{x<0\;\;or\;\;x>\frac14} $$

and on the other side

$$5-\frac2x<3\implies2\;\frac{x-1}x<0\iff \color{red}{0<x<1}$$

Take the common domain to both red expressions (why?) and you get

$$\frac14< x<1$$

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Hint:

Note that your denominator is defined iff $|5-\frac{2}{x}|\leq3$.

As denominator it is not allowed to equalize $0$ so you come to $|5-\frac{2}{x}|<3$.

This inequality must be solved to find the domain. You know how to start now.

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