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Tips and Tricks: Inequalities

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Re: Tips and Tricks: Inequalities [#permalink]

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New post 22 Sep 2013, 04:19
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]

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New post 22 Sep 2013, 06:04
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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]

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New post 03 May 2014, 06:25
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]

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New post 18 May 2014, 10:21
Do inequalities on the GMAT only refer to integers?

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Re: Tips and Tricks: Inequalities [#permalink]

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New post 19 May 2014, 02:27
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]

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New post 19 May 2014, 20:25
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]

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New post 11 Jun 2014, 04:01
Really helpful post.. Kudos +1...

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Re: Tips and Tricks: Inequalities [#permalink]

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New post 16 Jun 2015, 23:15
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Re: Tips and Tricks: Inequalities [#permalink]

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New post 07 Jun 2016, 00:10
Zarrolou wrote:
hb wrote:
Can you elaborate more on the external and internal values ? That should clear the concept completely for me. Is it possible for you to draw a shaded area on the graph that explains what you mean by the external and internal values as defined by x1 and x2 ?


Here are some more examples tips-and-tricks-inequalities-150873.html#p1225182

Say that you have \(x_1=3\) and \(x_2=5\)

External values:
------------------(3)-------------(5)------------------
Internal values:
------------(3)------------------(5)------------------

For x_1=-10 and x_2=-1
External values:
------------------(-10)-------------(-1)------------------
Internal values:
------------(-10)------------------(-1)------------------

External values=" values greater then the greatest root, and smaller than the smallest root".
Internal values="values in between the two roots".

In the image below there are two graphical examples. The first one represents the solution for
\(x^2-8x+15>0\)
The second one for
\(-x^2+8x-15>0\)

Hope everything is clear, let me know.
Thanks


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
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Re: Tips and Tricks: Inequalities [#permalink]

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Re: Tips and Tricks: Inequalities   [#permalink] 14 Jun 2017, 07:34

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