The given system of inequalities is
\(\frac{4x}{3}-\frac{9}{4}<x+\frac{3}{4}\) ...(i)
\(\frac{7x-1}{3}-\frac{7x+2}{6}>x\) ....(ii)
Now,
\(\frac{4x}{3}-\frac{9}{4}<x+\frac{3}{4}\)
⇒ \(\frac{16x-27}{12}<\frac{4x+3}{4}\)
⇒ 4(16x − 27) < 12(4x + 3)
⇒ 64x − 108 < 48x + 36
⇒ 64x − 48x < 36 + 108
⇒ 16x < 144
⇒ x < 9
So, the solution set for first inequation (i) is the interval (−∞, 9)
And \(\frac{7x-1}{3}-\frac{7x+2}{6}>x\)
⇒ \(\frac{2(7x-1)-(7x+2)}{6}>x\)
⇒ \(\frac{14x-2-7x-2}{6}>x\)
⇒ 7x − 4 > 6x
⇒ 7x − 6x > 4
⇒ x > 4
So, the solution set for inequation (ii) is the interval (4, ∞)
The solution set for inequations (i) and (ii) are graphed on the real line in fig (i) and fig (ii) respectively.
Hence the common solution for both inequalities is
x ∈ (4,9)
