Quadratic inequalities used the zeros to split the number line. The same sign-chart method works for any polynomial, and for fractions of polynomials, once you add the points where a denominator is 0. You already know how a polynomial's graph changes sign.
Idea clave
Move everything to one side and factor. Mark the zeros of the numerator and the zeros of the denominator on a number line. Test one number in each piece. Keep the pieces with the sign you want. Include numerator zeros only for $\ge$ or $\le$; never include a zero of a denominator.
Por qué funciona
Polynomials (and fractions of polynomials, away from their undefined points) are continuous: they can't change sign without passing through 0 or through a break. So the critical numbers are the only places a sign can change, and each piece between them has a single sign. Multiplying both sides by an expression is risky because if the expression is negative the inequality flips, and you don't know its sign in advance. A sign chart avoids that.
Worked examples
Ejemplo resuelto: A cubic
Solve $(x + 3)(x - 1)(x - 4) \ge 0$.
- Zeros: $-3$, $1$, $4$.
- Test: $x = -4$ gives $(-)(-)(-) < 0$; $x = 0$ gives $(+)(-)(-) > 0$; $x = 2$ gives $(+)(+)(-) < 0$; $x = 5$ gives $(+)(+)(+) > 0$.
- Keep the positive pieces and the zeros.
Respuesta: $[-3, 1] \cup [4, \infty)$
Ejemplo resuelto: A fraction
Solve $\frac{x - 2}{x + 1} \le 0$.
- Critical numbers: $2$ (numerator) and $-1$ (denominator).
- The quotient is negative between them and positive outside.
- Include $x = 2$ (the quotient is 0 there) but not $x = -1$ (undefined).
Respuesta: $(-1, 2]$
Error común
Multiplying both sides by $x + 1$ to clear the fraction. That gives $x - 2 \le 0$, so $x \le 2$, which wrongly includes $x = -5$: there $\frac{-7}{-4} > 0$. When $x + 1$ is negative, multiplying flips the inequality. Use a sign chart, and test a number from your answer in the original inequality.