Solve each of the following system of equations in R.

4*x* − 1 ≤ 0, 3 − 4*x* < 0

#### Solution

\[\text{ We have }, 4x - 1 \leq 0\]

\[ \Rightarrow 4x \leq 1\]

\[ \Rightarrow x \leq \frac{1}{4} (\text{ Dividing both the sides by } 4)\]

\[ \Rightarrow x \in ( - \infty , \frac{1}{4}] . . . (i)\]

\[\text{ Also }, 3 - 4x < 0\]

\[ \Rightarrow 0 > 3 - 4x\]

\[ \Rightarrow 4x > 3\]

\[ \Rightarrow x > \frac{3}{4} \text{ Dividing both sides by } 4\]

\[ \Rightarrow x \in \left( \frac{3}{4}, \infty \right) . . . (ii)\]

\[\text{ Hence, the solution of the given set of inequalities is the intersection of } (i) \text{ and } (ii) . \]

\[\text{ But }, \left( - \infty \frac{1}{4} \right) \cap \left( \frac{3}{4}, \infty \right) = \phi\]

\[\text{ Thus, the given set of inequations has no solution } .\]