3.7 Polynomial and Rational Inequalities

Move all terms to one side and solve $x^{4} - x^{3} - 13 x^{2} + 5 x + 12<4 x$. Factor completely over the reals and show a sign table. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

numerator factor $x - 4$, signs from left to right: negative, negative, negative, negative, positive.
numerator factor $x - 1$, signs from left to right: negative, negative, negative, positive, positive.
numerator factor $x + 1$, signs from left to right: negative, negative, positive, positive, positive.
numerator factor $x + 3$, signs from left to right: negative, positive, positive, positive, positive.
With zero on the right, factor the expression as $\left(x - 4\right) \left(x - 1\right) \left(x + 1\right) \left(x + 3\right)$.
Cut points: $-3$, $-1$, $1$, $4$. Original excluded inputs: none.
Sign chart: between $-\infty$ and $-3$: positive; between $-3$ and $-1$: negative; between $-1$ and $1$: positive; between $1$ and $4$: negative; between $4$ and $\infty$: positive.
Use open endpoints at zeros because the comparison is strict; poles are always excluded.
The solution is $\left(-3, -1\right) \cup \left(1, 4\right)$.
Written-work review:

• Show the stated algebra and preserve every excluded input.
• Include all requested graph, sign-table, model or explanation components.
• Compare the written work with the supplied reference solution; selecting a statement alone does not establish mastery.