Problem 103 - Algebra and Trigonometry

Question

Write a rational inequality whose solution set is \((-\infty,-4) \cup[3, \infty)\)

Step-by-Step Solution

Answer

The rational inequality whose solution set is $(-\infty,-4) \cup [3,\infty)$ is \(\frac{x+4}{x-3} < 0\) for x < -4 and \(\frac{x+4}{x-3} > 0\) for x ≥ 3.

Step 1: Set up the rational inequality

A rational inequality that would satisfy these conditions could be constructed from the factors (x+4) and (x-3), which have roots -4 and 3 respectively. That is, expression (x+4)/(x-3) would be greater than or equal to zero for x in the interval $[3,\infty)$ and less than zero for x in $(-\infty,-4)$.

Step 2: Test the inequality

\(x \leq -4\) then \(\frac{x+4}{x-3}<0\) and for \(x \geq 3\), \(\frac{x+4}{x-3}>0\). When x=-5, the expression is -1/8 which is less than 0. When x=4, the expression is 8 which is greater than 0.

Step 3: Formulate the final inequality

After confirming that the expression is fulfilling the needed conditions, the final inequality is determined. The rational inequality should be \(\frac{x+4}{x-3}> 0\) when x is inside the interval $[3,\infty)$ and less than zero \( \frac{x+4}{x-3}< 0\) when x is inside the interval $(-\infty,-4)$. This is the needed rational inequality for the solution set given