Question

x-2/x-3+x-4/x-5=10/3

Answer 1

The required answer is in the attachment.

Answer 2

$\Large{\underline{\underline{\bf{Solution:-}}}}$

$⟹\frac{x - 2}{x - 3} + \frac{x - 4}{x + 5} = \frac{10}{3} \\ \\⟹ \frac{(x - 2)(x - 5) + (x - 4)(x - 3)}{(x - 3)(x - 5)} = \frac{10}{3} \\ \\ ⟹ \frac{ ({x}^{2} - 7x + 10) + ( {x}^{2} - 7x + 12)}{( {x}^{2} - 8x + 15) } = \frac{10}{3} \\ \\⟹ \frac{(2 {x}^{2} - 14x + 22)}{ ({x}^{2} - 8x + 15) } = \frac{10}{3}$

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${\bold{\underline{\underline{By\: cross \: multiplication:-}}}}$

$⟹3(2 {x}^{2} - 14x + 22) = 10( {x}^{2} - 8x + 15) \\ \\ ⟹6 {x}^{2} - 42x + 66 = 10 {x}^{2} - 80x + 150 = 0 \\ \\⟹ 6 {x}^{2} - 10 {x}^{2} - 42x + 80x + 66 - 150 = 0 \\ \\ ⟹ - 4 {x}^{2} + 38x - 84 = 0$

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${\bold{\underline{\underline{By\: middle \: term \: splitting\: method:-}}}}$

$⟹- 4 {x}^{2} + 38x - 84 = 0 \\ \\⟹ - 2(2 {x}^{2} - 19x + 42) = 0 \\ \\⟹ 2 {x}^{2} - 7x - 12x + 42 = 0 \\ \\⟹ x(2x - 7) - 6(2x - 7) = 0 \\ \\ ⟹(2x - 7)(x - 6) = 0$

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${\bold{\underline{\underline{Case-1:-}}}}$

$⟹2x - 7 = 0 \\ \\ ⟹2x = 7 \\ \\⟹ x = \frac{7}{2}$

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${\bold{\underline{\underline{Case-2:-}}}}$

$⟹x - 6 = 0 \\ \\⟹ x = 6$

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$\Large{\underline{\underline{\bf{Thanks:-}}}}$