Here is an inequality question in the problem solving format

Which of the following is correct if x is a real number and (x - 11)(x - 3) is negative?

A. x^2 + 5x + 6 < 0

B. x^2 + 5x + 6 > 0

C. 5 - x < 0

D. x - 5 < 0

E. 11 - x > 0

The question states that (x - 11)(x - 3) < 0

The solution to this inequality can be obtained as follows:

Simplifying, we get x < 11 and x > 3. i.e., 3 < x < 11.

Simplifying, we get x > 11 and x < 3. This is an impossible solution as x cannot simultaneously be greater than 11 and less than 3. So, infeasible.

So, the solution set to the inequality is 3 < x < 11.

Now, let us look at the options.

x^2 + 5x + 6 < 0.

Factorizing the quadratic expression, we get (x + 2)(x + 3) < 0.

As we did with the expression in the question, we could get the solution set as -3 < x < -2. We know that the value of x as stated in the question lies between 3 and 11. So, this is not possible.

x^2 + 5x + 6 > 0.

The solution set will be exactly the opposite as that for choice A. so, x < -3 and x > -2. The question has values of x lying between 3 and 11, which satisfy the condition x > -2.

Hence, choice B is the correct answer.