The number of positive integral solutions of  [x² (4-3x)² (x-2)⁴ ] / [(x-5)⁵ . (2x-7)⁶] greater than equal to 0

a). 1
b). 2
c). 3
d). 4

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Please find below the solution to the asked query:

$$\begin{gathered} \frac{{\mathrm{x}}^2 {\left(4-3\mathrm{x}\right)}^2 {\left(\mathrm{x}-2\right)}^4}{{\left(\mathrm{x}-5\right)}^5 {\left(2\mathrm{x}-7\right)}^6}\ge 0...\left(\mathrm{i}\right) \\ {\left(2\mathrm{x}-7\right)}^6, {\mathrm{x}}^2, {\left(4-3\mathrm{x}\right)}^2, {\left(\mathrm{x}-2\right)}^4 \mathrm{are} \mathrm{always} \mathrm{positive} \mathrm{so} \mathrm{they} \mathrm{don}\text{'}\mathrm{t} \mathrm{effect} \mathrm{inequality} \\ \mathrm{except} \mathrm{when} \mathrm{they} \mathrm{are} 0. \\ {\mathrm{x}}^2, {\left(4-3\mathrm{x}\right)}^2, {\left(\mathrm{x}-2\right)}^4 \mathrm{are} \mathrm{in} \mathrm{numerator} \mathrm{and} \mathrm{we} \mathrm{want} \ge 0, \mathrm{hence} \mathrm{x}=0, \mathrm{x}=\frac{4}{3}, \mathrm{x}=2 \\ \mathrm{will} \mathrm{be} \mathrm{in} \mathrm{out} \mathrm{solution}. \\ {\left(2\mathrm{x}-7\right)}^6 \mathrm{is} \mathrm{in} \mathrm{denominator},\mathrm{hence} \mathrm{x}=\frac{7}{2} \mathrm{should} \mathrm{not} \mathrm{be} \mathrm{in} \mathrm{solution}. \\ \mathrm{Odd} \mathrm{power} \mathrm{can} \mathrm{be} \mathrm{written} \mathrm{in} \mathrm{their} \mathrm{linear} \mathrm{form}, \mathrm{Hence} \left(\mathrm{i}\right) \mathrm{becomes} \\ \frac{1}{\mathrm{x}-5}\ge 0 \\ \mathrm{x}>5 \\ \mathrm{Hence} \mathrm{complete} \mathrm{solution} \mathrm{is} \\ \mathrm{x}\in \left(5,\infty \right)\cup \left\{0,\frac{3}{4},2\right\}-\left\{\frac{7}{2}\right\} \\ \mathrm{As} \mathrm{you} \mathrm{can} \mathrm{see}, \mathrm{there} \mathrm{are} \mathrm{infinite} \mathrm{positive} \mathrm{integers} \mathrm{satisfying} \mathrm{the} \mathrm{inequality}. \end{gathered}$$

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