Answer
$$\frac{x-3}{x+4} \geq 0$$
Work Step by Step
To find the rational inequality, it helps to first re-write the intervals in the following manner: $x \lt-4$ and $x\geq 3$. In turn, we can also write it as $(x \lt 0 - 4)$ and $(x \geq 0 + 3)$ which is the same as writing $(x + 4) \lt 0$ and $(x - 3) \geq 0$. By drawing the inequality lines, we can see the relationship between each factor if they were to be combined into an inequality polynomial (see image). Both inequalities combine into positives, so we could write the polynomial as $$(x+4)(x-3) \geq 0$$ However, this initial inequality would include the value $x = 4$ which is not a part of the original solution set. Therefore, we can write it as an irrational inequality instead: $$\frac{x-3}{x+4} \geq 0$$ By having $x + 4$ in the denominator, we ensure that the value $x = 4$ cannot be included and, thus, we satisfy the solution set.
