Question

Solve each of the following system of equations in R. 

\[\frac{2x - 3}{4} - 2 \geq \frac{4x}{3} - 6, 2\left( 2x + 3 \right) < 6\left( x - 2 \right) + 10\]

Answer 1

\[\frac{2x - 3}{4} - 2 \geq \frac{4x}{3} - 6\]
\[ \Rightarrow \frac{2x - 3}{4} - \frac{4x}{3} \geq - 6 + 2\]
\[ \Rightarrow \frac{3\left( 2x - 3 \right) - 16x}{12} \geq - 4\]
\[ \Rightarrow 6x - 9 - 16x \geq - 48\]
\[ \Rightarrow - 10x \geq - 39\]
\[ \Rightarrow 10x \leq 39 \left[ \text{ Multiplying both sides by } - 1 \right]\]
\[ \Rightarrow x \leq \frac{39}{10}\]
\[ \Rightarrow x \in ( - \infty , \frac{39}{10}] . . . (i)\]
\[\text{ Also }, 2\left( 2x + 3 \right) < 6\left( x - 2 \right) + 10\]
\[ \Rightarrow 4x + 6 < 6x - 12 + 10\]
\[ \Rightarrow 4x + 6 < 6x - 2\]
\[ \Rightarrow 6x - 2 > 4x + 6\]
\[ \Rightarrow 6x - 4x > 6 + 2\]
\[ \Rightarrow 2x > 8\]
\[ \Rightarrow x > 4 \]
\[ \Rightarrow x \in \left( 4, \infty \right) . . . (ii)\]
\[\text{ Hence, thesolution of the given set of inequalities is theintersection of } (i) \text{ and } (ii), \]
\[( - \infty , \frac{39}{10}] \cap \left( 4, \infty \right) = \varnothing \]
\[ \text{ which is an empty set } . \]
\[\text{ Thus, there is no solution of the given set of inequations } . \]