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Tips and Tricks: Inequalities

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Re: Tips and Tricks: Inequalities [#permalink] New post 22 Sep 2013, 03:19
Narenn wrote:
TirthankarP wrote:
Bunuel wrote:


Zarrolou,

In this particular question we can see that there 4 possible ranges : x<-\frac{1}{3}, -\frac{1}{3}<x<0, 0<x<\frac{2}{5} and x>\frac{2}{5}

So how is your method applicable here?


15x - \frac{2}{x} > 1 -------> \frac{15x^2 - 2}{x} - 1 > 0 -----------> \frac{15x^2 - x - 2}{x} > 0 -----------> \frac{(3x+1)(5x-2)}{x} > 0

The Critical points for numerator -1/3, 2/5
Critical point for denominator 0

Total critical points ------------- -1/3 --------------- 0 --------------- 2/5 ---------------

Since the sign of original inequality is positive, the expression will be positive in the rightmost region and in other regions it will be alternatively negative and positive.

That means ---------------- -1/3 +++++++++ 0 ---------------- 2/5 ++++++++++++

Hence Solution of the inequality is -1/3 < x 0 , x > 2/5

Hope that helps! :)


If the sentence marked in red is a rule then what will happen if the inequality is negative?
And does this rule has any impact if the sign of the coefficient of x^2 changes (+15 in this case)?
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Re: Tips and Tricks: Inequalities [#permalink] New post 22 Sep 2013, 05:04
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TirthankarP wrote:
If the sentence marked in red is a rule then what will happen if the inequality is negative?


If the sign of the inequality is negative, then you have to go with negative interval. For more info refer these articles Part 1, Part 2

TirthankarP wrote:
And does this rule has any impact if the sign of the coefficient of x^2 changes (+15 in this case)?


In that case, you can move negative x^2 to other side of the inequality so as to make it positive and then can solve as per the earlier method.

Hope that helps! :)
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Re: Tips and Tricks: Inequalities [#permalink] New post 03 May 2014, 05:25
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?
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Re: Tips and Tricks: Inequalities [#permalink] New post 18 May 2014, 09:21
Do inequalities on the GMAT only refer to integers?
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Re: Tips and Tricks: Inequalities [#permalink] New post 19 May 2014, 01:27
Expert's post
SirBolly wrote:
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.
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Re: Tips and Tricks: Inequalities [#permalink] New post 19 May 2014, 19:25
anishasjkaul wrote:
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?



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!!!

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Re: Tips and Tricks: Inequalities [#permalink] New post 11 Jun 2014, 03:01
Really helpful post.. Kudos +1...
Re: Tips and Tricks: Inequalities   [#permalink] 11 Jun 2014, 03:01
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