Linear Programming Mock Test Linear Inequalities Question 16

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

Options:

A) infinite

B) two solutions.

C) three solutions

D) no solutions

Answer:

Correct Answer: A

Solution:

Let us consider the following example: $ 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\text{ 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},\text{i.e.},,x\in ( -\infty ,\frac{55}{10} ] $ Solution set is given by $ ( -\infty ,\frac{55}{10} ]\cap (4,\infty )=( 4,\frac{55}{10} ] $ Thus, the system has infinitely many solutions.