Question

solve for x:x-4/x-5 + x-6/x-7 = 10/3 ; x is not equal to 5 or 7

Answer 1

x = (54+10)/8 = 8
X = (54-10)/8 = 44/8 = 11/2

Answer 2

Answer :

Remember : $\bold{x \neq 5,7}$

Now,

$\bold{\dfrac{x - 4}{x - 5} + \dfrac{x - 6}{x - 7} = \dfrac{10}{3}}$

$\to \bold{\dfrac{(x - 4)(x - 7) + (x - 5)(x - 6)}{(x - 5)(x - 7)} = \dfrac{10}{3}}$

$\to \bold{3 \{(x - 4)(x - 7) + (x - 5)(x - 6) \} = 10(x - 5)(x - 7)}$

$\to \bold{3 ( {x}^{2} - 11x + 28 + {x}^{2} - 11x + 30) = 10( {x}^{2} - 12x + 35)}$

$\to \bold{3(2 {x}^{2} - 22x + 58) = 10( {x}^{2} - 12x + 35)}$

$\to \bold{6 {x}^{2} - 66x + 174 = 10 {x}^{2} - 120x + 350}$

$\to \bold{(10 - 6) {x}^{2} + ( - 120 + 66) x + (350 - 174) = 0}$

$\to \bold{4 {x}^{2} - 54x + 176 = 0}$

$\to \bold{2 {x}^{2} - 27 x + 88 = 0}$

$\to \bold{2 {x}^{2} - 16x - 11x + 88 = 0}$

$\to \bold{2x(x - 8) + - 11(x - 8) = 0}$

$\to \bold{(x - 8)(2x - 11) = 0}$

∴ either x - 8 = 0 or, 2x - 11 = 0

⇨ either x = 8 or, 2x = 11

⇨ either x = 8 or, x = $\bold{\dfrac{11}{2}}$

Hence, the required solution be

x = 8, $\bold{\dfrac{11}{2}}$

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