Given:
\(\cfrac{{\text{x}}-3}{{\text{x}} + 1} < 0 , x \in R\)
Signs of x – 3
x – 3 = 0 → x = 0 (Adding both the sides by 3)
x – 3 < 0 → x < 3 (Adding both the sides by 3)
x – 3 > 0 → x > 3 (Adding both the sides by 3)
Signs of x + 1
x + 1 = 0 → x = -1 (Subtracting both the sides by 1)
x + 1 < 0 → x < -1 (Subtracting both the sides by 1)
x + 1 > 0 → x > -1 (Subtracting both the sides by 1)
\(\cfrac{{\text{x}}-3}{{\text{x}} + 1} \) is not defined when x = -1
The interval that satisfies the condition that \(\cfrac{{\text{x}}-3}{{\text{x}} + 1} \) is -1 <x<3
Therefore
x ϵ (-1, 3)