The inequality sign is far more intimidating than it need be. In many cases, just think of the inequality sign as an equal sign. Which case that is will be described below – along with the instance in which you have to switch the sign. But with a little bit of practice, inequality should be the least of your worries on the GRE.

## The Basics

x – 3 < 0

Note how the sign is tapered on the left side. The smaller side of the inequality corresponds to the smaller value. In this case, x – 3 is less than zero.

The first step is to change the inequality sign into an equal sign.

x – 3 = 0

Next solve: x = 3

Now place the inequality sign in the same place it was: x < 3.

See, that wasn’t too bad. Now let’s introduce a new sign: the less than equal sign.

2x + 3 >= 4

Again, replace sign with an equal sign and solve:

2x + 3 = 4

2x = 1

x = 1/2

x >= 1/2

To interpret this mathematically speaking, we need a number line. In this little video I will show you how the number line relates to inequalities.

## The scary negative

But things aren’t always that straightforward. If the variable in an inequality is multiplied by a negative number (e.g. -2x), then we have to change the sign.

Let’s have a look:

-2x + 1 > 5.

Again repeat steps, but because we are dividing by a -2, at the very end we must flip the sign, so > becomes <.

-2x + 1 = 5

-2x = 4

x = -2

x< -2.

It’s always a good idea to test your answer. So let’s plug in a value less than -2, say -4, and see if it satisfies the original equation:

(-2)(-4) + 1 > 5

9 > 5.

Because 9 > 5, we clearly pointed the sign in the right direction.

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If this was a little basic for you, I will have an advanced inequality post coming soon.

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## Takeaway

The inequalities you see on the GRE will be pretty basic. As long as you understand the above, you should be ready for the test.

#### Special Note:

To find out where inequalities sit in the “big picture” of GRE Quant, and what other Quant concepts you should study, check out our post entitled:

What Kind of Math is on the GRE? Breakdown of Quant Concepts by Frequency

Hey chris,

Your method is right, but it requires approximation… you wont get a correct answer.

Though the answer is 65 as the upper limit of n would be less than 75 and lower limits greater than 45…

Great to know these aren’t pertinent to gre.

Thanks a ton…. i’ll go through the propounded article…..

Hi Aman,

Oh, actually you are right. It does say LEAST, so the answer is clearly (C). Hmm…I’m still not sure if the GRE would include this. Maybe, but I wouldn’t worry about it. BTW what was the original source?

Hi Chris ,

I have a doubt in inequalities….

Q)Which of the following option should be the least value of n that satisfy the inequality,

2^n>(10)^15?

a)30

b)45

c)60

d)75

e)90

I wanted to know methods to solve one like of this…?

Please help me out.

Ah yes, good question. I actually wrote a post that addressed these very questions!

https://magoosh.com/gre/2012/gre-exponents-practice-question-set/

They are indeed tricky – meaning check your answer to make sure you didn’t fall for any traps.

For your question above:

10^15 = 2^15 x 5^15. Then things become kind of nasty, without a calculator. But I think of it this way: 5^3 = (apprx) 2^7, thus 5^15 = (apprx) 2^35. Thus 2^35 x 2^15 = 10^15, so n is apprx 50, which is not an answer choice.

My guess is this question is faulty. (A) there is no correct answer (B) you may see this on the GMAT but it is beyond the scope of the GRE.

The questions in the link above are a more accurate indication of what you will see test day :).

Hope that helps

Brilliant simplification to this difficult problem.

Thanks, Chris. 🙂

what’s the answer? Is it option C — 60?

2^n > 10^15

> (2*5)^15

> (2^15) * (5^15)

> (2^15) * (5^3)^5

Lets look at the options now

a)30 2^30 ==> (2^15) * (2^15)

==> (2^15) * (2^3)^5

(2^3)^5 < (5^3)^5 # 8^5 (2^15) * (2^30)

==> (2^15) * (2^6)^5

(2^6)^5 < (5^3)^5 # 64^5 (2^15) * (2^45)

==> (2^15) * (2^7)^5

(2^7)^5 > (5^3)^5 # 128^5 > 125^5

So n=60 is the least value of n that satisfy the above inequality.

d)75 can be ignored.

e)90 can be ignored.