The solution set of a(x - 1)/(x - 2) > 2 is A, and 3 is not in A. Find the range of a.

Step-by-step solution

1. 1

   Resist solving the inequality in general. Determining $A$ for every $a$ would need a sign chart with cases on the sign of $a$ and on the pole at $x=2$. But the question asks only about a single point, so evaluate there instead — that is the entire shortcut.

2. 2

   Check that $x=3$ is admissible. The expression is undefined only at $x=2$. Since $3\ne 2$, the point $x=3$ lies in the domain, so "$3\notin A$" genuinely means the inequality fails at $x=3$ rather than being undefined there.

3. 3

   Substitute $x=3$.

   $\frac{a(3-1)}{3-2}=\frac{2a}{1}=2a$

   so membership of $3$ in $A$ is exactly the statement $2a>2$.

4. 4

   Negate the membership condition. $3\notin A$ means the strict inequality does not hold:

   $2a>2\ \text{is false}\iff 2a\le 2\iff a\le 1$

   The negation of a strict inequality includes equality, so $a=1$ is allowed: it makes the left side exactly $2$, which is not greater than $2$.

5. 5

   Verify at the boundary and just outside it. With $a=1$: $\tfrac{1(3-1)}{3-2}=2$, and $2>2$ is false, so $3\notin A\;\checkmark$. With $a=1.5$: the value is $3>2$, so $3\in A$ — correctly excluded. With $a=0$: the value is $0>2$, false, so $3\notin A\;\checkmark$. Hence the range is $a\le 1$.