<s>[INST] How can I solve this inequality? $ \frac{x+14|x|-10}{|4x-6|-21}>3$ First I looked the x that doesnt belong to this function.
$$|4x -6| - 21 \neq 0$$
$$ x \neq \frac{-15}{4}$$ and $$ x \neq \frac{27}{4}$$
Then I found the roots of the x
$$x = 0$$
$$x = \frac{3}{2}$$
After I found the roots I wrote the inequality like this:
$$ x + 14|x| - 10 > 3|4x-6| - 63$$
$$ x + 14|x| > 3|4x-6| - 53$$
to finish we may write differents function for each values of x based on roots we found.
for $$ x<0$$ the function is $$ x - 14x > 3(6-4x) - 53$$
for $$x<\frac{3}{2}$$ we have $$ x + 14x > 3(6-4x) - 53$$ for $$x \geq \frac{3}{2}$$ the function is $$ x + 14x > 3(4x-6) - 53$$
But using this inequalities I couldnt find the solutions for x!! If we calculate that on wolfram we can see the solutions for x are $$ x < \frac{-15}{4}$$ and $$ x > \frac{27}{4}$$. Can anyone explain me why?
 [/INST] Transform the inequality:
$$\frac{x+14|x|-10-3|4x-6|+63}{|4x-6|-21}>0.$$
Consider the $3$ cases:
$$1) \ \begin{cases} x<0 \\ \frac{x-35}{4x+15}>0\end{cases} \Rightarrow \begin{cases} x<0 \\ x<-\frac{15}{4} \ \ \text{or} \ \  x>35\end{cases} \Rightarrow \color{blue}{x<-\frac{15}{4}}.$$
$$2) \ \begin{cases} 0<x<\frac32 \\ \frac{27x+35}{-4x-15}>0\end{cases} \Rightarrow \begin{cases} 0<x<\frac32 \\ -\frac{15}{4}<x<-\frac{35}{27}\end{cases} \Rightarrow \emptyset.$$
$$3) \ \begin{cases} x>\frac32 \\ \frac{3x+71}{4x-27}>0 \end{cases} \Rightarrow \begin{cases} x>\frac32 \\ x<-\frac{71}{3} \ \ \text{or} \ \ x>\frac{27}{4}\end{cases} \Rightarrow \color{blue}{x>\frac{27}{4}}.$$
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