GMAT Inequalities, Quadratic Equation

Inequalities is a big favorite of the GMAT test makers. Inequalities, especially presented as a data sufficiency question that appear in the GMAT test are many a times potential land mines.

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:

Case 1: When (x – 11) < 0 and (x - 3) > 0, the product will be negative.
Simplifying, we get x < 11 and x > 3. i.e., 3 < x < 11. Case 2: When (x – 11) > 0 and (x – 3) < 0, the product will be negative
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. Choice (A)
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. Choice (B)
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.