Question

solve the inequality \\(\frac{x + 2}{x - 5} < 0\\). enter the solution in interval notation.

Explanation:

Step1: Analyze the sign of the fraction

A fraction \(\frac{a}{b}\) is negative when \(a\) and \(b\) have opposite signs. So we consider two cases:
Case 1: \(x + 2>0\) and \(x - 5<0\)
Case 2: \(x + 2<0\) and \(x - 5>0\)

Step2: Solve Case 1

For \(x + 2>0\), we get \(x>- 2\).
For \(x - 5<0\), we get \(x<5\).
The intersection of \(x>-2\) and \(x < 5\) is \(-2

Step3: Solve Case 2

For \(x + 2<0\), we get \(x<-2\).
For \(x - 5>0\), we get \(x>5\).
There is no intersection for \(x<-2\) and \(x > 5\), so this case has no solution.

Answer:

\((-2, 5)\)