<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] Your way is good, first consider


*

*$|4x-6|-21> 0$


and we have


*

*$x<\frac32\implies -4x+6-21> 0 \implies x<-\frac {15} 4$

*$x\ge\frac32\implies 4x-27> 0 \implies x>\frac {27} 4$


then


*

*$|4x-6|-21> 0$ for $x<-\frac {15} 4$ and $x>\frac {27} 4$

*$|4x-6|-21< 0$ for $-\frac {15} 4<x<\frac {27} 4$

*$|4x-6|-21= 0$ for $x=-\frac {15} 4$ and $x=\frac {27} 4$


Then consider the cases

1) $-\frac {15} 4<x<\frac {27} 4$



*

*$\frac{x+14|x|-10}{ |4x-6|-21 }>3\implies x+14|x|-10<3|4x-6|-63$


and


*

*$x<0 \implies -13x-10<3(-4x+6)-63$

*$0\le x<\frac32 \implies 15x-10<3(-4x+6)-63$

*$x\ge\frac32 \implies 15x-10<3(4x-6)-63$



2) $x<-\frac {15} 4$ and $x>\frac {27} 4$



*

*$\frac{x+14|x|-10}{ |4x-6|-21 }>3\implies x+14|x|-10>3|4x-6|-63$


and


*

*$x<-\frac {15} 4 \implies -13x-10<3(-4x+6)-63$

*$x>\frac {27} 4 \implies 15x-10<3(4x-6)-63$

</s>