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Tips and Tricks: Inequalities

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Re: Tips and Tricks: Inequalities [#permalink]
Do inequalities on the GMAT only refer to integers?
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Re: Tips and Tricks: Inequalities [#permalink]
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.

Theory on Inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html
everything-is-less-than-zero-108884.html
graphic-approach-to-problems-with-inequalities-68037.html

All DS Inequalities Problems to practice: search.php?search_id=tag&tag_id=184
All PS Inequalities Problems to practice: search.php?search_id=tag&tag_id=189

700+ Inequalities problems: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope this helps.
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Re: Tips and Tricks: Inequalities [#permalink]
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!!!

Kudos if you like!!!
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Re: Tips and Tricks: Inequalities [#permalink]
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Re: Tips and Tricks: Inequalities [#permalink]
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.
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Zarrolou wrote:
connexion wrote:
Hi Zarrolou ,
Thanks for your tip. Can you please elaborate it with more examples? I have difficulty in understanding it.

Thanks.

Sure!
Here are all possible cases:
1)$$x^2-3x+2>0$$
2)$$x^2-3x+2<0$$

The roots of $$x^2-3x+2=0$$ are x=1 and x=2
In 1 the sign of x^2 is + and the operator is > => they are "the same" so we take the external values
$$x<1$$ and $$x>2$$
In 2 the signs are not the same => we take the internal values
$$1<x<2$$

in these the sign of x^2 is -
3)$$-2x^2+3x+2>0$$
4)$$-2x^2+3x+2<0$$

The roots of $$-2x^2+3x+2 =0$$ are $$x=2$$ and $$x=-\frac{1}{2}$$.
In 3 the sign of x^2 is - and the operator is > => the are not the same so we take the internal values
$$-\frac{1}{2}<x<2$$
In 4 they are the same (-,<) external values
$$x<-\frac{1}{2}$$ and $$x>2$$

Hope it's clear now. Let me know

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?
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gmatprep001 wrote:
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?

Hey gmatprep001 ,

Let me solve the equation for you:

$$-2x^2+3x+2 =0$$ can be written as $$2x^2-3x-2 =0$$.

Now, $$2x^2-3x-2 =0$$

==> $$2x^2 - 4x + x - 2 =0$$

==> $$2x(x - 2) + 1 (x - 2) =0$$

==> either $$2x + 1 = 0$$ or $$(x - 2) =0$$

==> $$x = -1/2$$ or x = 2.

Does that make sense?
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Re: Tips and Tricks: Inequalities [#permalink]
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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Re: Tips and Tricks: Inequalities [#permalink]
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