Problem 103 - College Algebra

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 \( \frac{x+4}{x-3}\geq 0\)

Step 1: Identify the Critical Points

The critical points from this problem are -4 and 3. They split the number line into the intervals \( (-\infty,-4), (-4,3), and (3, \infty) \). Identify the critical points and understand their importance in formulating the inequality.

Step 2: Formulate the Inequality

Form an inequality where these critical numbers are the roots of a rational function. A sensible option to realise this is: \( \frac{x+4}{x-3}\geq 0 \). For this function, x = -4 and x = 3 are the roots. Next step would be to solve this inequality and compare its solution with the given one.

Step 3: Solve the Inequality

Solve the inequality \( \frac{x+4}{x-3}\geq 0 \). The set solution for this inequality matches the given intervals which are \(-\infty,-4) \cup [3, \infty) \). This confirmed that \( \frac{x+4}{x-3}\geq 0 \) is the correct inequality.