Question

what are the critical points for the inequality x^2-4/x^2-5x+6<0?

Answer 1

X=-2, x=2, and x=3 these are the critical points

Answer 2

Answer:

The critical points are 2,-2 and -3.          

Step-by-step explanation:

Given the inequality

$\frac{x^2-4}{x^2-5x+6}<0$

we have to find the critical points for the inequality

We reduce the given inequality into factored form

$a^2-b^2=(a-b)(a+b)$

$x^2-4=x^2-2^2=(x-2)(x+2)$

$x^2-5x+6=x^2-3x-2x+6=x(x-3)-2(x-3)=(x-2)(x-3)$

The inequality becomes

$\frac{x^2-4}{x^2-5x+6}<0$

$\frac{(x-2)(x+2)}{(x-2)(x-3)}<0$

Critical points are those values of domain where it is not differentiable or its derivative is 0 or we can say the values at which the numerator and denominator  is equal to zero.

Hence, the critical points are 2,-2 and -3.