Determine all real numbers $x $ that satisfy the inequality

Question

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.

Answer 1

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?