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Chapter 3: Problem 103

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

Short Answer

Expert verified
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 鈮� 3.

Step by step solution

01

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 $(-\infty,-4)$.
02

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.
03

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

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