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I want to ask about way of solving exponential inequalities, I am going to show you two similar examples, but their solving is kinda different.

First example: $3^{2x}-10\cdot3^x+9>0$ $(3^x-9)(3^x-1)>0$ $3^x\in(-\infty;1)\cup(9;+\infty)$ So to find x we need to solve:

$3^x>9$ and $3^x>1$ (1)

and we get: $x\in(-\infty;0)\cup(2;+\infty)$

Second example: $5^{2x}-6\cdot5^x+5<0$ $(5^x-5)(5^x-1)<0$ $5^x\in(1;5)$ So to find x we need to solve:

$\begin{cases}5^x<5\\5^x>1\end{cases}$ (2)

and we get: $x\in(0;1)$

My question is why in first example I get answer by solving 2 equations (1) seperately but in second example I get answer by solving system of equations (2)

2 Answers 2

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When you have $ab \gt 0$, you can have $a\gt 0, b \gt 0$ or $a \lt 0, b \lt 0$. So for your first example you should have $[3^x \gt 9$ and $3^x \gt 1]$ or $[3^x \lt 9$ and $3^x \lt 1]$. You can combine each of the square brackets because one of the inequalities is always true when the other one is to get $3^x \gt 9$ or $3^x \lt 1$ (note the or, not and), leading to the solution you have. For the second, if $ab \lt 0$ one of $a$ and $b$ is greater than zero and the other is less. So you can have $[5^x \gt 5$ and $5^x \lt 1]$ or $[5^x \lt 5$ and $5^x \gt 1]$. In this case the inequalities in the first bracket are inconsistent, so that one can be ignored and we get the inequalities you cite.

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There is no difference: in the first case you should solve $ (3^x>9\ \mbox{and}\ 3^x>1)\ \mbox{or}\ (3^x<9\ \mbox{and}\ 3^x<1), $ and in the second case you should solve $ (5^x>5\ \mbox{and}\ 5^x<1)\ \mbox{or}\ (5^x<5\ \mbox{and}\ 5^x>1). $ It just happens that $(3^x>9$ and $3^x>1)$ reduces to $(3^x>9)$, that $(3^x<9$ and $3^x<1)$ reduces to $(3^x<1)$, and that $(5^x>5$ and $5^x<1)$ is impossible.

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    @user9325: following your post, I did proceed to edit the answer. I then saw Didier's response; I guess I'm hesitant to step in and alter others' answers before addressing the issue directly with the answerer? I don't want to come across as disrespectful. But I appreciate your comment. At this point, anyway, my edits need to "pass muster" with the Mods before they "stick", anyway.2011-05-04