$\frac{x^{2} - 2 x + 3}{x^{2} - 4 x + 3} > - 3$

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Solution

$(\frac{x^{2} - 2 x + 3}{x^{2} - 4 x + 3}) > - 3 o r (\frac{x^{2} - 2 x + 3}{x^{2} - 4 x + 3}) + 3 > 0 o r (\frac{x^{2} - 2 x + 3 + 3 x^{2} - 12 x + 9}{x^{2} - 4 x + 3}) > 0 o r (\frac{4 x^{2} - 14 x + 12}{(x - 3) (x - 1)}) > 0 o r (\frac{2 x^{2} - 7 x + 6}{(x - 3) (x - 1)}) > 0 o r (\frac{2 x^{2} - 4 x - 3 x + 6}{(x - 3) (x - 1)}) > 0 o r (\frac{2 x (x - 2) - 3 (x - 2)}{(x - 3) (x - 1)}) > 0 o r (\frac{(x - 2) (2 x - 3)}{(x - 3) (x - 1)}) > 0 ∴ x \in (- \infty, 1) \cup ((\frac{3}{2}), 2) \cup (3, \infty)$

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