Question

Write the set of values of x satisfying the inequations 5x + 2 < 3x + 8 and \[\frac{x + 2}{x - 1} < 4\] 

Answer 1

\[\text{ We have }, \]
\[5x + 2 < 3x + 8 \text{ and } \frac{x + 2}{x - 1} < 4\]
\[ \Rightarrow 2x < 6 \text{ and } \frac{x + 2}{x - 1} - 4 < 0\]
\[ \Rightarrow x < 3 \text{ and } \frac{x + 2 - 4x + 4}{x - 1} < 0\]
\[ \Rightarrow x \in ( - \infty , 3) \text{ and } \frac{- 3x + 6}{x - 1} < 0\]
\[ \Rightarrow x \in ( - \infty , 3) \text{ and } \frac{- x + 2}{x - 1} < 0\]
\[\text{ For } \frac{- x + 2}{x - 1} < 0, \text{ critical points are 1 and } 2 . \]
\[ \Rightarrow x \in (2, \infty ) \cup ( - \infty , 1)\]
\[ \therefore x \in ( - \infty , 1) \cup (2, 3)\]