Find all values of $a$ for which the inequality $(a - 1) x^{2} - (a + 1) x + a + 1 > 0$ is satisfied for all real $x$ .

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Solution

$(a - 1) x^{2} - (a + 1) x + a + 1 > 0$

For this inequality to hold

$(a + 1)^{2} - 4 (a + 1) (a - 1) < 0$

$(a + 1) (a + 1 - 4 a + 4) < 0$

$(a + 1) (3 a - 5) > 0 ⟹ a \in (- \infty, - 1) \cup (\frac{5}{3}, \infty)$

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