Question

Solve each of the following system of equations in R. 

$$\frac{2x-3}{4}-2\ge \frac{4x}{3}-6,2(2x+3)<6(x-2)+10$$

Answer 1

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