Factors always of the same sign

When a factor has constant sign (a parabola with $\Delta<0$, a sum of squares, a constant) it does not contribute to changing the sign of the product: it can be simplified, remembering to reverse the sense if the factor is negative.

Example — Numerator or denominator always of the same sign

Solve $\dfrac{-x^2+x-1}{x+3}\ge 0$.

Analysis of the numerator: $-x^2+x-1$. We compute $\Delta = 1-4 = -3 < 0$ and observe $a=-1<0$. By the rule of the six cases (case $a<0,\,\Delta<0$), the parabola lies always below the $x$-axis: the numerator is always negative.

We can then “divide” both sides by the numerator (negative), reversing the sense: $\frac{1}{x+3} \le 0 \;\Longleftrightarrow\; x+3 < 0 \;\Longleftrightarrow\; \boxed{x < -3}$ (Note that $x=-3$ must be excluded because it makes the denominator zero.)

Summary of the technique: when a factor is “always $+$” or “always $-$” (parabolas with $\Delta<0$, sums of squares, constants…), it can be simplified: only the other factors actually contribute to the sign.

Links

Topics: Inequalities
Concepts: Discriminant · Fractional inequality · Sign analysis
Skills: Solving inequalities · Simplifying