Solve each of the following system of equations in R.
Question:

Solve each of the following system of equations in R.

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

Solution:

$\frac{2 x-3}{4}-2 \geq \frac{4 x}{3}-6$

$\Rightarrow \frac{2 x-3}{4}-\frac{4 x}{3} \geq-6+2$

$\Rightarrow \frac{3(2 x-3)-16 x}{12} \geq-4$

$\Rightarrow 6 x-9-16 x \geq-48$

$\Rightarrow-10 x \geq-39$

$\Rightarrow 10 x \leq 39 \quad$ [Multiplying both sides by $-1]$

$\Rightarrow x \leq \frac{39}{10}$

$\Rightarrow x \in\left(-\infty, \frac{39}{10}\right] \quad \ldots(\mathrm{i})$

Also, $\quad 2(2 x+3)<6(x-2)+10$

$\Rightarrow 4 x+6<6 x-12+10$

$\Rightarrow 4 x+6<6 x-2$

$\Rightarrow 6 x-2>4 x+6$

$\Rightarrow 6 x-4 x>6+2$

$\Rightarrow 2 x>8$

$\Rightarrow x>4$

$\Rightarrow x \in(4, \infty) \quad \ldots$ (ii)

Hence, the solution of the given set of inequalities is the intersection of (i) and (ii),

$\left(-\infty, \frac{39}{10}\right] \cap(4, \infty)=\emptyset$

which is an empty set.

Thus, there is no solution of the given set of inequations.

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