Question

(1/x+4 ) - (1/x-7) = 11/30 , x is not equal to -4 and 7

Answer 1

1/x+4-1/x-7=11/30
taking LCM and simplifying  
x-7-x-4/(x^2-3x-28)=11/30
-11/x^2-3x-28=11/30
x^2-3x-28=-30
x^2-3x+2=0
(x-2)(x-1)=0
x=2 or 1

Answer 2

$\huge\underline\mathbb{Holamate}$

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

$⟹\frac{(x - 7) - (x + 4)}{(x + 4)(x - 7)} = \frac{11}{30 } \\ \\ ⟹ \frac{x - 7 - x - 4}{ {x}^{2} - 4x - 7x - 28} = \frac{11}{30} \\ \\ ⟹ \frac{ - 11}{ {x}^{2} - 3x - 28 } = \frac{11}{30}$

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

$⟹ - 11(30) = 11( {x}^{2} - 3x - 28)$

${\bold{\underline{\underline{-11 \:will \:be \: cut \: by \:11}}}}$

$⟹ - 30 = {x}^{2} - 3x - 28 \\ \\ ⟹{x}^{2} - 3x - 28 + 30 = 0 \\ \\⟹{x}^{2} - 3x + 2 = 0$

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${\bold{\underline{\underline{By\: Quadratic\: formula:-}}}}$

$⟹{x}^{2} - 3x + 2$

➦Here, a = 1 , b = -3 and c = 2

$</strong><strong>⟹</strong><strong>{b}^{2} - 4ac = 0 \\ \\</strong><strong>⟹</strong><strong> { (- 3)}^{2} (4)(1)(2) = 0 \\ \\</strong><strong>⟹</strong><strong> 9 - 8 = 0 \\ \\ </strong><strong>⟹</strong><strong>1 ≻0$

➦This means they have two different real roots.

$⟹x = \frac{ \sqrt{ - b} ∓ \sqrt{ {b - 4ac}^{2} } }{2a} \\ \\ ⟹x = \frac{3∓ \sqrt{1} }{2 \times 1} \\ \\⟹ x = \frac{3 ∓ \: 1}{2}$

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

$⟹x = \frac{3 + 1}{2} \\ \\ ⟹x = \frac{4}{2} \\ \\⟹ x = 2$

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

$⟹x = \frac{3 - 1}{2} \\ \\⟹ x = \frac{2}{2} \\ \\⟹ x = 1$

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$\huge\underline\mathbb{Thanks}$