Find all values of $a$ for which the inequality $(a - 3) x^{2} - 2 a x + 3 a - 6 < 0$ is satisfied for all values of $x$ .

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Solution

$(a - 3) x^{2} - 2 a x + 3 a - 6 < 0 \forall x ϵ R$
$a - 3 < 0$
$a < 3$
$a ϵ (- \infty, 3)$
$b < 0$
$(2 a)^{2} - 4.3 (a - 3) (a - 2) < 0$
$4 a^{2} - 4.3 (a^{2} - 5 a + 6) < 0$
$a^{2} - 3 a^{2} + 15 a - 18 < 0$
$2 a^{2} - 15 a + 18 > 0$
$2 a^{2} - 12 a - 3 a + 18 > 0$
$(2 a - 3) (a - 6) > 0$
$a ϵ (- \infty, \frac{3}{2}) \cup (6, \infty)$
$a ϵ (- \infty, \frac{3}{2})$

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