Question

Solve each of the following system of equations in R. 

\[\frac{7x - 1}{2} < - 3, \frac{3x + 8}{5} + 11 < 0\]

Answer 1

\[\frac{7x - 1}{2} < - 3\]
\[ \Rightarrow 7x - 1 < - 6\]
\[ \Rightarrow 7x < - 6 + 1\]
\[ \Rightarrow x < \frac{- 5}{7}\]
\[ \Rightarrow x \in \left( - \infty , \frac{- 5}{7} \right) . . . (i)\]
\[\text{ Also }, \frac{3x + 8}{5} + 11 < 0\]
\[ \Rightarrow \frac{3x + 8 + 55}{5} < 0\]
\[ \Rightarrow 3x + 63 < 0\]
\[ \Rightarrow 3x < - 63\]
\[ \Rightarrow x < - 21 \]
\[ \Rightarrow x \in \left( - \infty , - 21 \right) . . . (ii)\]
\[\text{ Hence, the solution to the given set of inequations is the intersection of } (i) \text{ and } (ii) . \]
\[\left( - \infty , \frac{- 5}{7} \right) \cap \left( - \infty - 21 \right) = \left( - \infty , - 21 \right)\]
\[\text{ Hence, the solution to the given set of inequations is } \left( - \infty , - 21 \right) .\]