(x+6)/(x^2+1)<0
1) Let's solve in ℝ: x + 6x2 + 1 < 0
We study the sign of the numerator
⇥The expression x + 6 is zero at x = -6, it is negative before and positive after
We study the sign of the denominator, which must not be zero
Let's study the sign of x2 + 1 by solving in ℝ: x2 + 1 < 0
In ℝ, an even power is never negative: x2 < -1 has no solution
Solution: x ∈ ∅
⇥So the expression x2 + 1 is always positive or zero
2) Let's draw up the sign table:
x-∞ -6 +∞
x + 6-0+
x2 + 1++
x + 6x2 + 1-0+
▶Solution obtained: x < -6