Solve the rational inequality $\frac{x^2-4}{x^2-5x+6}\ge 0$

Step 1 ：First, factorize the numerator and denominator: $\frac{(x-2)(x+2)}{(x-2)(x-3)}\ge 0$

Step 2 ：Next, find the critical points by setting the numerator and denominator equal to zero: Critical points are $x=-2,2,3$

Step 3 ：Then, test the intervals $(-\mathrm{\infty },-2)$, $(-2,2)$, $(2,3)$, and $(3,\mathrm{\infty })$ by choosing a test point from each interval and evaluating the sign of $\frac{(x-2)(x+2)}{(x-2)(x-3)}$

Step 4 ：For $x=-3$, $x=0$, $x=2.5$, and $x=4$, we get +, -, +, and - respectively.

Step 5 ：Finally, the solution to the inequality $\frac{x^2-4}{x^2-5x+6}\ge 0$ is the union of the intervals where the value of the rational expression is positive, as well as the points where the expression is zero.