Sometimes, Quantitative Comparison (Quant Comp) and Algebra may turn out to be a deadly combination on the GRE. Not so if you are good at handling inequalities! After all Quant Comp questions are nothing but inequality questions.

Quant Comp questions ask you whether Quantity A is always **>** Quantity B, or whether Quantity B is always **>** Quantity A, or whether Quantity A is always **=** Quantity B. If the answer to any of these questions is yes, you have got your answer – (A), (B), and (C) respectively! If the answer to any of the above questions is “may be or may not be”, even then you have got your answer! In that case, your answer is (D)!

Let’s discuss just two simple inequality related properties and see how that can help us in solving Quant Comp questions involving Algebraic expressions.

**Inequality Property 1**: In any inequality you can add or subtract same quantity on both the sides and the inequality still holds good.

For example,

If *x* > *y*, where *x*, *y*, and *a* are real numbers,

*x* +*a *> *y+ a *and

*x* − *a *> *y – a*

It does not matter whether *x*, *y*, and *a* are positive or negative.

**Inequality Property 2**: In any inequality you can multiply or divide both the sides by a positive number and the inequality still holds good. However, if you multiply or divide both the sides by a negative number the inequality gets reversed.

If *x* > *y*, where *x*, *y*, and *a* are real numbers,

*ax* > a*y *and x/a > y/a if *a* is positive, but

*ax* < a*y *and x/a< y/a if *a* is negative.

It doesn’t matter whether *x *and *y* are positive or negative.

Now, let’s see how knowing these properties can help us solve some GRE Quant Comp questions.

**Example 1 A:**

x is a real number

Quantity A Quantity B

4*x* − 7 *2x+3
*

As per Inequality Property 1 above, in any inequality, you can add or subtract same quantity on both the sides and the inequality still holds good.

Let us subtract 2*x* from and add 7 to both the quantities.

We get

Quantity A Quantity B

2x 10

As per Inequality Property 2, in any inequality you can multiply or divide both the sides by a positive number and the inequality still holds good.

Thus by dividing both sides by 2, we get

Quantity A Quantity B

x 5

Therefore, the answer is (D)

Now, let’s look at a small variation.

**Example 1B:**

x is a real number greater than 5

Quantity A Quantity B

4x-7 2x+3

The same way, this will lead us to

Quantity A Quantity B

x 5

Since *x* is a real number greater than 5, the answer becomes (A)

Here’s another variation.

**Example 1C:
**

x is a real number greater than 5

Quantity A Quantity B

4x-8 2x+3

This time, we subtract 2*x* from and add 8 to both the quantities and divide both the quantities by 2.

We get

Quantity A Quantity B

x 5.5

We are told that *x* is greater than 5. Obviously, *x* is not necessarily an integer. Therefore, x may be less than, equal to, or greater than 5.5, and therefore the answer is (D).

In Part II of this blog we will explore the Inequality Property 2 that we mentioned of in the beginning.

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