How to know what to plug in?

Yes... We just have to recognize the pattern. Also in this question it would help to notice that with every increase in x, the LHS is increasing by a factor of 4 whereas, RHS is only increasing by a factor of 3. of course, that is only a restatement of my previous post.

Kudos Please... If my post helped.

how long can we go on pluggin in numbers..

This will come through practice.

The good thing is - there are n number of ways to solve a question. the bad thing is - you need to find the fastest. There is no need to plug a 100 numbers in this case. You just need to find whether the situation holds true for every positive x or not. Your first step should always be to reduce the number of ' numbers you need to plug in' .

First, I'd suggest you put an '=' instead of the </> sign. so essentially 4^x = 3^x * 3.

This is one way of quickly finding what number to plug. Find the value of x that satisfies the equation. This is your critical point. Use a few values to the left of this and a few to the right.

so is 4^x<3^x*3
or (4/3)^x<3 (because 3^x is always positive you can bring it to the LHS)
or (1.33)^x<3

for 1 it is true for 10000000 it is not. hence, the statement is insufficient.

MacFauz's method is good as well.

There are two ways you can use:

1. Use logic/algebra instead of plugging in numbers. It's tough to prove something using numbers. It's easier to disprove.

Is 4^x < 3^(x+1)?
Is (4/3)^x < 3?

4/3 is greater than 1. When you raise it to a high power, it will take a big enough value. There is no reason it should stay less than 3. When you raise 1 to a power, it stays 1. When you raise a number less than 1 to a positive integral power, the number becomes even lesser. These are some number properties you need to work through and be comfortable with.

2. Look for the transition point.

Find out where it will be equal. Then the behavior will be different on either side of the transition point (as done in the post above)

I have discussed these two methods in another question. Check it out here: how-to-stop-searching-for-values-to-prove-a-statement-127787.html#p1047350

Remember, you cannot shut your mind and just plug in numbers and get your answer. If it were so easy, everyone would have scored Q51. The point is that you need to always keep your mind on alert and use logic even if you are using number plugging.