असमानता \(\frac{1}{5}\left( {\frac{{3x}}{5} + 4} \right) \ge \frac{1}{3}(x - 6)\) का हल लिखिए। 

  1. \(x \le \frac{{105}}{8}\)
  2. \(x \ge \frac{{105}}{8}\)
  3. \(x \ge 120\)
  4. \(x \le 120\)

Answer (Detailed Solution Below)

Option 1 : \(x \le \frac{{105}}{8}\)
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Detailed Solution

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संकल्पना:

दिया गया है \(\frac{1}{5}\left( {\frac{{3x}}{5} + 4} \right) \ge \frac{1}{3}(x - 6)\)

\(\begin{array}{l} \Rightarrow 3\left( {\frac{{3x}}{5} + 4} \right) \ge 5(x - 6)\\ \Rightarrow \left( {\frac{{9x}}{5} + 12} \right) \ge 5x - 30\\ \Rightarrow \left( {30 + 12} \right) \ge - \frac{{9x}}{5} + 5x\\ \Rightarrow 42 \ge \frac{{ - 9x + 25x}}{5}\\ \Rightarrow 42 \ge \frac{{16x}}{5}\\ \Rightarrow \frac{{42 \times 5}}{{16}} \ge x\\ \Rightarrow x \le \frac{{105}}{8} \end{array}\)

अतः विकल्प (1) सही उत्तर है। 

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