Question

Solve for inequality:- (1/x-5 + 1/x+5 + 1/x-7 + 1/x+7) > 0

Answer 1

$</p><p> \sf \rightarrow \quad \left( \frac{1}{x - 5} + \frac{1}{x + 5} + \frac{1}{x - 7} + \frac{1}{x + 7} \right) \gt 0 \\ \\ \sf \rightarrow \quad \left( \frac{x \cancel{ + 5} + x \cancel{- 5}}{ ( x + 5)(x - 5) } + \frac{x \cancel{+ 7} + x \cancel{- 7}}{(x + 7)(x - 7)} \right) \gt 0 \\ \\ \sf \rightarrow \quad \frac{2x( {x}^{2} - {7}^{2} ) + 2x( {x}^{2} - {5}^{2} ) }{(x + 5)(x - 5)(x + 7)(x - 7)} \gt 0 \\ \\ \sf \rightarrow \quad \frac{2x( {x}^{2} - 25 + {x}^{2} - 49) }{(x + 5)(x - 5)(x + 7)(x - 7)} \gt 0 \\ \\ \sf \rightarrow \quad \frac{2x(2 {x}^{2} - 24) }{(x + 5)(x - 5)(x + 7)(x - 7)} \gt 0 \\ \\ \sf \rightarrow \quad \frac{4x( {x}^{2} - 12) }{(x + 5)(x - 5)(x + 7)(x - 7)} \gt 0 \\ \\ \sf \rightarrow \quad \frac{4x (x + 2 \sqrt{3})(x - 2 \sqrt{3}) }{(x + 5)(x - 5)(x + 7)(x - 7)} \gt 0</p><p>$

$</p><p> \sf Critical \: points: 5, -5, 7, -7, 0, 2\sqrt{3} , - 2\sqrt{3} \\ </p><p>$

$</p><p> \sf By \: the \: wavy \: curve \: method , \: we \: get </p><p></p><p>$

$</p><p> \sf x \in \{ - \infty, \: - 7 \} \: \cup \: \{ - 5, \: -2\sqrt{3} \} \: \cup \: \{ 0, \: 2\sqrt{3} \} \: \cup \: \{ 5, \: \infty \}</p><p>$