Let us look at a different dimension of the concept that we discussed in Part I.

**Example 2A:**

x is a real number greater than 5

__Quantity A__ __Quantity B__

1/4x-7 1/2x+3

Even if we want to plug in numbers for *x*, it may not be very convenient to do so with the question in its present form. A good idea is to multiply both sides by (4*x *– 7) (2*x* + 3)

We get

x is a real number greater than 5

Quantity A __Quantity B__

2x+3 4x-7

Use the same process as used in Examples 1A, 1B, and 1C, and you will get the answer (B).

*The most important point to note here is that we are able to do this multiplication because we know that (4x – 7)(2x + 3) is always positive as both (4x – 7) and (2x + 3) are always positive.*

Let’s look at a variation of this question.

**Example 2B:
**

x is a real number greater than 1

Quantity A __Quantity B__

1/4x-7 1/2x+3

*In this case, it will be erroneous to multiply both sides by (4x – 7) (2x + 3) as, although (2x + 3) is always positive, (4x – 7) can be either positive or negative, and, therefore, (4x – 7) (2x + 3) can be either positive or negative!*

However, there’s still a way to simplify this to some extent. We can multiply both sides by (2*x*+3), and we get

x is a real number greater than 1

Quantity A __Quantity B__

2x+3/4x-7 1

Now it may be a good idea to plug in values for *x*.

Quantity A __Quantity B__

2x+3/4x-7 1

Put *x = *1.1 5.2/-2.6

Quantity B is greater; cancel out (A) and (C).

Quantity A __Quantity B__

Put *x = *2 7/1 1

Quantity A is greater; eliminate (B) and your answer is (D).

Let’s look at another variation of this question.

**Example 2C:**

x is a real number less than − 5

Quantity A __Quantity B__

1/4x-7 1/2x+3

*Now, we note that both **(4x – 7) and (2x + 3) are always negative and, therefore, (4x – 7)(2x + 3) is always positive. We can just multiply both sides by (4x – 7)(2x + 3) and follow the same process. *

We get

x is a real number less than − 5

Quantity A __Quantity B__

2x+3 4x-7

Use the same process as used in Examples 1A, 1B, and 1C.

This time, however, your answer will be (A) as *x* is a real number less than − 5

Here is one more variation — this time with a twist!

**Example 2D:**

x is a real number less than 7 but greater than −3

Quantity A __Quantity B__

1/x-7 1/x+3

Hey! Don’t just start putting values for *x* or start manipulating the expressions algebraically. This time you really don’t have to do anything. You just have to figure out that (*x*–7) is always negative and (*x*+3) is always positive. Consequently, your answer is (B). This one is really an easy question. You don’t have to do anything more.

As we say at Manya-The Princeton Review, “*Work smarter, not harder.*”

In subsequent Parts, we will discuss quant comp questions involving more than one variable. See you: till then.