## The SBC GMAT Files

Inequalities
Staff, Test Prep New York/Test Prep San Francisco

The Quantitative section of the GMAT is not as much about testing your ability to perform lengthy calculations as it is about testing your knowledge of underlying math principles. On the GMAT, you can often see questions that use small and convenient integer values, yet solving such questions may be very hard, especially if you forget some basic rules and properties. One particularly tricky type of question deals with inequalities. So let’s take a closer look at them.
Inequalities are very similar to equations, and can be treated the same way as equations but with one major exception.
You can add or subtract numbers to both sides of an inequality, keeping the inequality true:
2 < 5; add 5 to both sides to get 7 < 10, which is still true.
2 < 5; subtract 5 from both sides to get -3 < 0, which is also true.

You can multiply or divide both sides of the inequality by a positive number, keeping the inequality true:
2 < 5; multiply both sides by 5 to get 10 < 25, which is true.
2 < 5; divide both sides by 5 to get 0.4 < 1, which is also true.

The only thing that is different, and is often tested on the GMAT, is that whenever you multiply or divide the inequality by a negative number, the inequality sign changes:

2 < 5; multiply both sides by -5 to get -10 < -25, but this inequality is wrong unless you flip the inequality sign and make it -10 > -25. You are still allowed to multiply and divide inequalities by negative numbers, but whenever you do so, do not forget to flip the inequality sign.
Let’s see how we can apply the rules above to solve a relatively easy Data Sufficiency question:
Is y > 3?

(1) 3y + 2 â‰¥ 8
(2) 12 â‰¤ 3y

A) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient.

One way to answer the question would be to find the exact value of y, but since neither statement provides such opportunity, let’s look whether we can narrow down the range of possible values of y to the point when we will be able to answer the question.
By subtracting 2 from both sides of the inequality in statement (1), we can rewrite it as 3y â‰¥ 6, dividing both sides of this inequality by 3, we get y â‰¥ 2, but this range includes values that are greater than 3, for example 3.5, and values that are less than three, for example 2.5. We cannot prove correctness or incorrectness of the inequality y â‰¥ 3. Statement (1) alone is insufficient, and we can eliminate choices A and D, which presume its sufficiency.
By dividing the inequality from statement (2) by 3, we get 4 â‰¤ y or y â‰¥ 4. The minimum possible value of y is 4, and it must be true that y >3.  The correct answer is B.
This was an easy warm-up question. Now that you better understand how manipulating inequalities can help you answer GMAT questions, let’s look at a somewhat harder example:
Which of the following inequalities is equivalent to ,ð‘Ž-ð‘‘. > ,ð‘-ð‘‘. if dâ‰ 0?
A) a > c
B) c > a
C) ,ð‘Žâˆ’ð‘-ð‘‘. > 0
D) ,ð‘Žâˆ’ð‘-ð‘‘. < 0
E) c < d
The easiest way to answer this question is to manipulate the inequality ,ð‘Ž-ð‘‘. > ,ð‘-ð‘‘. until you get an inequality from one of the answer choices. A tempting but incorrect way to do so, would be to multiply both sides of the inequality by d, to get:

d Ã—  ,ð‘Ž-ð‘‘. > d Ã— ,ð‘-ð‘‘. or a > c

The problem with this approach is that we do not know whether d is positive or negative. If d is positive, than a > c, but if d is negative, then the opposite is true and c > a.
Choices C and D have 0 on the right side, so let’s compare our inequality to zero:
,ð‘Ž-ð‘‘. > ,ð‘-ð‘‘.; subtract ,ð‘-ð‘‘. from both sides to get ,ð‘Ž-ð‘‘. – ,ð‘-ð‘‘. > 0, which is equivalent to ,ð‘Žâˆ’ð‘-ð‘‘. > 0.
A bit more complex type of inequality you may come across on the GMAT is inequalities with absolute value. Again, if you understand the underlying principles of absolute values and inequalities, these should not be hard.
The first thing you need to do is to isolate the expression with absolute value on one side of the inequality sign. This can be achieved by manipulating the inequality as we have discussed above.
Once you have isolated the absolute value expression, you have to apply a different strategy, depending on whether the absolute value expression is greater or less than the expression to which it is compared.
CASE 1: Absolute value expression is GREATER than the expression to which it is compared.
The general rule for this case can be written as: If |x| > y, then x > y OR x < -y
For example:
|x+2|>6
x+2 > 6 OR x+2 < -6
x > 4 OR x < -8
So the inequality |x+2|>6, will be true for all values of x that are either greater than 4 or less than -8. Graphically we can show this like so:

CASE 2: Absolute value expression is LESS than the expression to which it is compared.
The general rule for this case can be written as: If |x| < y, then x < y AND x > -y.
For example:
|x+2|< 6
x +2 <6 AND x+2 > -6
x<4 AND x>-8
The inequality |x+2|< 6, will be true for all value of x that are less than 4 but greater than -8. We can show this graphically like so:

There are several main points you should take away from this post:
–        You can manipulate inequalities the same way as you do with equations, but do not forget to flip the inequality sign when you divide or multiply the inequality by a negative number.
–        When you divide or multiply an inequality by a variable, consider two cases: a variable is positive and the sign of the inequality remains unchanged; or a variable is negative and the sign of the inequality must be reversed.
–        When you deal with absolute value inequalities, isolate the absolute value expression and then use one of the two rules above, depending on whether the absolute value expression is greater or less than the expression to which it is compared.
Inequalities are sometimes used by the GMAT writers to test your understanding of number properties. Here’s a data sufficiency style example
Is ,ð‘¥-5. > ð‘¦ ?

1)0 < ð‘¦ < 1.
2)The absolute value of ð‘¥ is greater than 1.

When approaching this kind of question, it can often be helpful to plug in numbers for the variables. Try plugging in each of the following values: 0, 1, -1, a positive number greater than 1, a positive number less than 1, a negative number less than -1, and a negative number greater than -1.
In the example above, we first evaluate statement 1 alone. Plugging in ð‘¥=0, we get ð‘¦>ð‘¥, plugging in ð‘¥=1, we get ð‘¦<ð‘¥. So, statement 1 alone is insufficient. Statement 2 tells us nothing about ð‘¦, so is clearly insufficient by itself. Evaluating both statements together, we can exclude all values of x other than positive numbers greater than 1 and negative values less than -1. Let’s try 2 and -2. ,2-5.=32, and 32>ð‘¦. ,âˆ’2-5. =âˆ’32, and âˆ’32<ð‘¦. Statements 1 and 2 together are insufficient. The answer is E.
Good Luck!

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