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Let $ r$ be a fixed real number. Determine all real numbers $x$ that satisfy the inequality $\frac{1}{1+x^2}$ $≤ r$ .

Can someone help me start this question? I am aware of a method when $r = 0$ , but I don't know how to do this since $r$ is unknown.

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HINT

Note that, since $1+x^2>0$, for $r\le 0$ there are not solutions then we can consider the case $r>0$ and obtain $$\frac{1}{1+x^2}\le r \iff (1+x^2)\frac{1}{1+x^2}\le (1+x^2)r \iff rx^2+r-1\ge 0$$

Can you proceed form here?

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  • $\begingroup$ I still don't understand where i would go from here, would you be able to give me one more hint for the next step following this? Having a complete mind blank. @gimusi $\endgroup$ – Taylor Nov 15 '18 at 17:42
  • $\begingroup$ @Taylor Just isolate the $x^2$ term. Show your work editing the OP, I'll take a look to it giving some more suggestion if you'll need it. $\endgroup$ – gimusi Nov 15 '18 at 17:48
  • $\begingroup$ Another way to isolate $x^2$ is by inverting both sides of the inequality: $1+x^2 \ge \frac{1}{r}$. This is allowed as long as $r \ne 0$. $\endgroup$ – Cuspy Code Nov 15 '18 at 17:53

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