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Question
\[\frac{x}{5} < \frac{3x - 2}{4} - \frac{5x - 3}{5}\]
Solution
\[\frac{x}{5} < \frac{3x - 2}{4} - \frac{5x - 3}{5}\]
\[ \Rightarrow 20 \times \left( \frac{x}{5} \right) < 20 \times \left( \frac{3x - 2}{4} - \frac{5x - 3}{5} \right) \left( \text{ Multiplying both the sides by 20 } \right)\]
\[ \Rightarrow 4x < 5\left( 3x - 2 \right) - 4\left( 5x - 3 \right)\]
\[ \Rightarrow 4x < 15x - 10 - 20x + 12\]
\[ \Rightarrow 4x < - 5x + 2\]
\[ \Rightarrow 4x + 5x < 2 (\text{ Transposing - 5x to the LHS) } \]
\[ \Rightarrow 9x < 2\]
\[ \Rightarrow x < \frac{2}{9} (\text{ Dividing both the sides by 9 }) \hspace{0.167em} \]
\[\text{ Hence, the solution set of the given inequation is } \left( - \infty , \frac{2}{9} \right) .\]
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