Tips and Tricks: Inequalities

Hi, Bunuel

The trick for inequalities written in the first post of this link, is it always true??... Will it work for all inequalities? Even third degree equation of X ( for example) with an inequality sign?

Do inequalities on the GMAT only refer to integers?

No. You cannot assume that a variable represents an integer unless you are explicitly told about it or unless it's obvious (for example when a variable represents the number of people, books, etc).

I suggest you to go through the following post ALL YOU NEED FOR QUANT.



Hope this helps.

Yes, it is always true. But a more easy approach is as follows:

Lets have an inequality of degree n of the form

(x-1)(x-2)(-3-x)(x-4)<=0,

Now here, we see that coefficient of highest power of x is negative. My approach is as follows:

Step 1: Make coefficient of highest power positive by changing the sign of inequality

(x-1)(x-2)(x+3)(x-4)>=0

Step 2: Identify critical point.

Four critical points are there -3,1,2,4

Since you have already made coefficient of highest power of x positive in first step, the inequality if always be positive beyond the right most critical point i.e. beyond 4 it is always positive

Step3: Plot the regions.

Now the +ve and -ve regions will alternate


>4 it is positive
2<x<4 it is negative
1<x<2 it is positive
-3<x<1 it is negative
x<-3 it is positive.

In this way you can solve the inequality.

In case we have an even power of a term as shown below:

(x-1)(x-2)^2(x-3)(x-4)>=0

The regions at the critical point 2 will not alternate. We will again start from right most critical point:

x>4 is positive
3<x<4 is negative
2<x<3 is positive
1<x<2 is positive
x<1 is negative

In case of an odd power the signs alternate normally.

Hope it makes sense!!!

Kudos if you like!!!

Really helpful post.. Kudos +1...

Really amazing post. Although this might sound a basic question to ask, l'm not quite sure when we have to take internal values. ls it when x is negative? Could someone please clarify why we need to consider internal values on second graph? Would be a great help.

Regards.
Ranaazad

I know it's a quite old post, however let me ask one question:
Are the solution for \(-2x^2+3x+2 =0\) not \(x=-2\), i.e. negative 2, and \(x=\frac{1}{2}\), i.e. positive 1/2, hence the solution range for \(-2x^2+3x+2>0\) is -2 < x < 1/2?

Abhimahna & gmatprep001

Even if you do not rewrite the equation, in original form also, you will get the same answer, How gmatprep001 arrived at x=-2?

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