It goes without saying that the toughest GMAT Quant Problems are GMAT Data Sufficiency questions involving Inequalities. One specific issue that arises when solving tougher questions of this type is how to combine the two statements when both involve inequalities. Let us use two GMAT Data Sufficiency questions to understand how to go about combining inequalities.

DIRECTIONS: Each data sufficiency problem below consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you are to choose option

A if statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
B if statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
C if BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient;
D if EACH statement ALONE is sufficient to answer the question asked;
E if statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

If x, y & z are positive integers then does z lie between x and y?

(1) x < 2z < y
(2) 2x < z < 2y

In an earlier post we had discussed how to visualize problems using a number line. The first key to open up this problem easily is to visualize it on a number line.

Statement I can be visualized in the following ways:

CASE 1: 0          x   2z                     y

CASE 2: 0         x                       2z y

In CASE 1, 2z lies close to x, z will then lie to the left of x and hence not between x and y.

In CASE 2, 2z lies close to y, in this case  z will lie between x and y.

So using statement 1 we cannot definitively conclude the position of z with respect to x and y.

Statement II can also be visualized similarly

CASE 1: 0         2x z                    2y

CASE 2: 0         2x                   z 2y

In CASE 1, z lies close to 2x, if 2x & 2y become x and y, z will still lie between them.

In CASE 2, z lies close to 2y, y will lie to the left of z and hence z will not be between x and y

How does one combine the two statements?

Combining two inequalities means using the information in both together — you have two statements and you can manipulate both pieces of information whichever way you want to to make further deductions. So in effect you can add, subtract, multiply or divide the two inequalities.

In this case, just a simple addition will do the job. Adding both statements will give you 3x < 3z < 3y or  x < z < y, so z does lie between x and y.

If xyz > 0, then is x > 0?

(1) xy > 0
(2) xz > 0

Statements I & II individually cannot conclusively tell you whether x > 0 since in both cases it can be either positive or negative.

How does one combine them?

Method 1: Multiply them — xy*xz > 0 or x(xyz) > 0, then x has to be positive since xyz > 0 and x(xyz)> 0

Method 2: Divide them — y/z > 0 which means that y & z are either both positive or both negative, so either ways yz has to be positive. If yz is positive and x(yz) is positive then x has to be positive.

So when you have to combine two statements, both of which are inequalities, do not think that you have to keep them as they are. You are free to manipulate them to deduce anything you can to answer the question posed.

Just remember, at each step do not forget the information given in the question. Always take the information given in the question along with the statement I individually, II individually and with both.

In the next Quantitative post we will see how even trivial information such as n is a positive integer in a DS question stem can be absolutely critical towards cracking Data Sufficiency problems.