<s>[INST] Inequality solving How must I proceed to solve inequalities of the following form:
$$-1≤r-1/r≤1$$ 
This is the final step for another problem that I'm attempting to solve and thus far, this is where I've gotten:
By taking the L.C.M and multiplying both sides by $r$ (given to take strictly positive (or zero) values), I got-
$$-r≤r²-2≤2r$$
Taking the left inequality separately, I found that $$(r-1)(r+2) \geq 0$$
which gives me 
$$r \in (-\infty, -2] \cup [1, \infty)$$
(Clearly, the only valid region is $[1,\infty)$, since $r$ can only take strictly positive/zero values but let me leave that there until the very end, making a mental note of it.)
The right half of the inequality gives me $r²-2r-2≤0$ and I have no insights as to how to solve this. 
Please help! Thanks in advance :) Regards. 
 [/INST] Let's consider two cases, $r > 0$ and $r < 0$.
Case 1
Suppose $r > 0$. Multiplying both sides of the inequality by $r$,
$$-r \leq r^2-1 \leq r.$$
Rearranging the left-side inequality,
$$r^2+r-1 \geq 0.$$
Using the quadratic equation to find the roots of $r^2+r-1$,
$$r = \frac{-1 \pm \sqrt{1^2-4(1)(-1)}}{2(1)} = \frac{-1 \pm \sqrt{5}}{2} .$$
Thus, our inequality becomes,
$$\left(r-\frac{-1 + \sqrt{5}}{2}\right)\left(r-\frac{-1 - \sqrt{5}}{2}\right) \geq 0.$$
From here, we get $r \in (-\infty, \frac{-1 - \sqrt{5}}{2}]$ or $r \in [\frac{-1 + \sqrt{5}}{2}, \infty).$ However, we assumed $r > 0$, thus we only have $r \in [\frac{-1 + \sqrt{5}}{2}, \infty).$
Now let's rearrange the right hand side of the inequality to get
$$r^2-r-1 \leq 0.$$
The roots of $r^2-r-1$ are,
$$\frac{-(-1) \pm \sqrt{(-1)^2-4(1)(-1)}}{2(1)} = \frac{1 \pm \sqrt{5}}{2}.$$
Thus, the right hand side inequality becomes,
$$\left(r-\frac{1 + \sqrt{5}}{2}\right)\left(r-\frac{1 - \sqrt{5}}{2}\right) \leq 0.$$
From here, we get $r \in [\frac{1 - \sqrt{5}}{2}, \frac{1 + \sqrt{5}}{2}]$. But remember $r > 0$, so this means $r \in (0, \frac{1 + \sqrt{5}}{2}]$.
Case 2
Let's multiply both sides of the original inequality by $r$, but remember to flip the direction of the inequalities,
$$-r \geq r^2 - 1 \geq r.$$
Rearranging the left and right side inequalities,
$$r^2+r-1 \leq 0 \text{ and } r^2-r-1 \geq 0.$$
These inequalties are the same as before except with the direction of the inequalties switched around so I'm not going to go through the same level as detail as in case 1.
For the left-hand inequality, we end up with 
$r \in [\frac{-1 - \sqrt{5}}{2}, \frac{-1 + \sqrt{5}}{2}]$; however, since $r < 0$, this becomes $r \in [\frac{-1 - \sqrt{5}}{2}, 0)$. For the right-hand inequality, we end up with $r \in (-\infty, \frac{1 - \sqrt{5}}{2}]$ or $r \in [\frac{1 + \sqrt{5}}{2}, \infty).$ However; since $r < 0$, this becomes just $r \in (-\infty, \frac{1 - \sqrt{5}}{2}].$
Putting everything together
Looking at the result of everything, we have,
$$\left(r \in [\frac{-1 + \sqrt{5}}{2}, \infty) \text{ AND } r \in [0, \frac{1 + \sqrt{5}}{2}] \right) \text{ OR } \left( r \in [\frac{-1 - \sqrt{5}}{2}, 0) \text{ AND } r \in (-\infty, \frac{1 - \sqrt{5}}{2}] \right).$$
More succintly,
$$r \in \left([\frac{-1 + \sqrt{5}}{2}, \infty) \cap [0, \frac{1 + \sqrt{5}}{2}] \right) \cup \left([\frac{-1 - \sqrt{5}}{2}, 0) \cap (-\infty, \frac{1 - \sqrt{5}}{2}] \right)$$
$$= [\frac{-1 + \sqrt{5}}{2},\frac{1 + \sqrt{5}}{2}] \cup [\frac{-1 - \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}]$$
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