Solve the given inequality for real x:
Question:

Solve the given inequality for real x$\frac{x}{4}<\frac{(5 x-2)}{3}-\frac{(7 x-3)}{5}$

Solution:

$\frac{x}{4}<\frac{(5 x-2)}{3}-\frac{(7 x-3)}{5}$

$\Rightarrow \frac{x}{4}<\frac{5(5 x-2)-3(7 x-3)}{15}$

$\Rightarrow \frac{x}{4}<\frac{25 x-10-21 x+9}{15}$

$\Rightarrow \frac{x}{4}<\frac{4 x-1}{15}$

$\Rightarrow 15 x<4(4 x-1)$

$\Rightarrow 15 x<16 x-4$

$\Rightarrow 4<16 x-15 x$

$\Rightarrow 4<x$

Thus, all real numbers $x$, which are greater than 4, are the solutions of the given inequality.

Hence, the solution set of the given inequality is $(4, \infty)$.

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