Given:
\(\cfrac{5{\text{x}} + 8}{4-{\text{x}}}<2, {\text{x}} \in R\)
Subtracting both the sides by 2
\(\cfrac{5{\text{x}} + 8}{4-{\text{x}}}\) - 2< 2 - 2
Signs of x:
x = 0
x < 0
x > 0
Signs of 4 – x:
4 – x = 0 → x = 4
(Subtracting 4 from both the sides, then dividing by -1 on both the sides)
4 – x < 0 → x > 4
(Subtracting 4 from both the sides, then multiplying by -1 on both the sides)
4 – x > 0 → x < 4
(Subtracting 4 from both the sides, then multiplying by -1 on both the sides)
At x = 4, \(\frac{{\text{x}}}{4-{\text{x}}}\) is not defined
Intervals satisfying the condition: < 0
x < 0 or x > 4
Therefore,
x є (-∞, 0) υ (4, ∞)