How to solve $−1 < (x+1)/(x+2) < 4$

Question

Solve $−1 < (x+1)/(x+2) < 4$

I know we can't multiply by the denominator because it might flip the inequality symbols, so how do I find $x$?

Answer 1

As suggested in the comment by conditionalMethod, consider two cases:

(1) $x+2>0$. Then $-1(x+2)<x+1<4(x+2),$ so $-3<2x$ and $-7<3x$.

If $x>-2$, $x>-\frac32$, and $x>-\frac73$, what can you say about $x$?

(2) $x+2<0$. Then $-1(x+2)\color{red}>x+1\color{red}>4(x+2),$ so $-3>2x$ and $-7>3x$.

If $x<-2$, $x<-\frac32$, and $x<-\frac73$, what can you say about $x$?