[tex] \frac{a}{ \frac{1}{2} } = \frac{b}{ \frac{1}{3} } \geqslant 2a = 3b \\ \\ \frac{3a + 2b}{5a - 3b} = \frac{3 \times \frac{3b}{2} + 2b}{5 \times \frac{3b}{2} - 3b } = \\ = \frac{ \frac{9b}{2} + 2b }{ \frac{15b}{2} - 3b} \\ = \frac{ \frac{9b + 4b}{2} }{ \frac{15b - 6b}{2} } \\ = \frac{13b}{9b } \\ = \frac{13}{9} [/tex]
