<s>[INST] Finding all $x$ for $\frac{2x - 13}{2x + 3} \lt \frac{15}{x}$ I'm trying to find all $x$ for the inequality $\frac{2x - 13}{2x + 3} \lt \frac{15}{x}$. In order to do this, I want to factor one side so that I can find all values where $x$ determines the term to equal $0$.
$$\frac{2x - 13}{2x + 3} \lt \frac{15}{x} \iff \frac{x(2x - 13) - 15(2x+3)}{x(2x + 3)} \lt 0$$
$$\iff \frac{2x^2 - 13x - 30x - 45}{2x(x + \frac{3}{2})} \lt 0 \iff \frac{x^2 - \frac{43}{2}x - \frac{45}{2}x}{x(x + \frac{3}{2})} \lt 0 $$
$$\iff \frac{x^2 - \frac{43}{2}x - \frac{45}{2}x}{x(x + \frac{3}{2})} \lt 0 \iff \frac{(x - \frac{43}{4})^2 - \frac{1939}{4}}{x(x+\frac{3}{2})} \lt 0$$
I don't know how to get any further, and I'm starting to get too high values to handle.
The next step as I can see would be to find an $x$ that makes $(x - \frac{43}{4})^2 = \frac{1939}{4}$. This, alongside the obvious ones for $x$ and $x+\frac{3}{2}$ (creating division by $0$), would help me find the possible values for $x$. 
But how do I get the last step? Or am I already dead wrong?
 [/INST] There is at least one incorrect "solution" above. 
The usual trick is to multiply both sides by something positive. So multiply both sides by $x^2(2x+3)^2$. Next, eliminate common factors in the numerators and denominators, bring everything over to the left hand side and then factorise. The inequality reduces to 
$x(2x-45)(2x+3)(x+1) < 0 \, .$
You need to find the values of $x$ for which an odd number of $x$, $2x-45$, $2x+3$ and $x+1$ are negative. So clearly $x = 0$, $x = 45/2$, $x = -3/2$ and $x = -1$ play an important role: these are the values for which the factors change sign. The final result, contrary to Iuli's "solution", is: $-3/2 < x < -1$ or $0 < x < 45/2$.
To see why Iuli's "solution" is incorrect, let $x = -1/2 \in (-3/2,45/2) \backslash \{0\}$, yet
$-7 \not< -30 \, .  $
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