JEE Main & Advanced Mathematics Linear Programming Question Bank Mock Test - Linear Inequalities

The system \[2(2x+3)-10<6(x-2)\] and \[\frac{2x-3}{4}+\ge \frac{2+4x}{3}\] has

A) infinite

B) two solutions

C) three sollutionns

D) no solutions

Correct Answer: A

Solution :

[a] Let

\[2(2x+3)-10<6(x-2)~\ldots (1)\]

\[\frac{2x-3}{4}+6\ge \frac{2+4x}{3}~\ldots (2)\]

\[(1)\Rightarrow 4x+6-10-6x+12<0\]

\[\Rightarrow -2x+8<0\]

\[\Rightarrow -2x<-8\Rightarrow x>4\,\,i.e.,x\in (4,\infty )\]

\[(2)\,\,\Rightarrow \frac{2x-3+24}{4}\ge \frac{2+4x}{3}\]

\[\Rightarrow 6x+63\ge 8+16x\]

\[\Rightarrow 6x-16\ge 8-63\]

\[\Rightarrow -10x\ge -55\]

\[\Rightarrow x\le \frac{55}{10}i.e.,x\in \left( -\infty ,\frac{55}{10} \right]\]

Solution set is given by \[\left( -\infty ,\frac{55}{10} \right]\cap (4,\infty )=\left( 4,\frac{55}{10} \right]\]

Thus, the system has infinitely solution.