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# If x is an integer, what is the value of x? (1) x^2 - 4x + 3

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If x is an integer, what is the value of x? (1) x^2 - 4x + 3 [#permalink]

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23 May 2010, 00:31
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If x is an integer, what is the value of x?

(1) x^2 - 4x + 3 < 0
(2) x^2 + 4x +3 > 0
[Reveal] Spoiler: OA

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press kudos, if you like the explanation, appreciate the effort or encourage people to respond.

Last edited by Bunuel on 04 Aug 2012, 00:59, edited 1 time in total.

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Re: x2 - 4x + 3 < 0 [#permalink]

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30 May 2010, 09:31
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Hussain15 wrote:
Bunuel wrote:
dimitri92 wrote:
If x is an integer, what is the value of x?

(1) x2 - 4x + 3 < 0
(2) x2 + 4x +3 > 0

Given: $$x=integer$$.

(1) $$x^2-4x+3<0$$ --> $$1<x<3$$--> as $$x$$ is an integer then $$x=2$$. Sufficient.

(2) $$x^2+4x+3>0$$ --> $$x<-3$$ or $$x>-1$$ --> multiple values are possible for integer $$x$$. Not sufficient.

With reference to your statement $$1<x<3$$ above, I calculated the range as $$x<1$$ or $$x<3$$
What I did wrong?

How to solve quadratic inequalities - Graphic approach.

$$x^2-4x+3<0$$ is the graph of parabola and it look likes this:
Attachment:

en.plot (1).png [ 3.79 KiB | Viewed 58603 times ]

Intersection points are the roots of the equation $$x^2-4x+3=0$$, which are $$x_1=1$$ and $$x_2=3$$. "<" sign means in which range of $$x$$ the graph is below x-axis. Answer is $$1<x<3$$ (between the roots).

If the sign were ">": $$x^2-4x+3>0$$. First find the roots ($$x_1=1$$ and $$x_2=3$$). ">" sign means in which range of $$x$$ the graph is above x-axis. Answer is $$x<1$$ and $$x>3$$ (to the left of the smaller root and to the right of the bigger root).

This approach works for any quadratic inequality. For example: $$-x^2-x+12>0$$, first rewrite this as $$x^2+x-12<0$$ (so that the coefficient of x^2 to be positive. It's possible to solve without rewriting, but easier to master one specific pattern).

$$x^2+x-12<0$$. Roots are $$x_1=-4$$ and $$x_1=3$$ --> below ("<") the x-axis is the range for $$-4<x<3$$ (between the roots).

Again if it were $$x^2+x-12>0$$, then the answer would be $$x<-4$$ and $$x>3$$ (to the left of the smaller root and to the right of the bigger root).

Hope it helps.
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Re: x2 - 4x + 3 < 0 [#permalink]

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23 May 2010, 01:46
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dimitri92 wrote:
If x is an integer, what is the value of x?

(1) x2 - 4x + 3 < 0
(2) x2 + 4x +3 > 0

Given: $$x=integer$$.

(1) $$x^2-4x+3<0$$ --> $$1<x<3$$ --> as $$x$$ is an integer then $$x=2$$. Sufficient.

(2) $$x^2+4x+3>0$$ --> $$x<-3$$ or $$x>-1$$ --> multiple values are possible for integer $$x$$. Not sufficient.

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Re: If x is an integer, what is the value of x? (1) x^2 - 4x + 3 [#permalink]

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24 May 2012, 07:54
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Hi,

I won't mind drawing the parabola for a quadratic equation. But a better approach is to check the sign of expression on a number line.

for example: (x-1)(x-2)(x-3)(x-7) < 0

To check the intervals in which this inequality holds true, we need to pick only one value from the number line.
Lets say x = 10, then (9)(8)(7)(3) > 0, in every alternate interval the sign would be + for the above expression

---(+)-----1--(-)--2---(+)--3----(-)-------7----(+)------

Thus, inequality would hold true in the intervals:
1 < x < 2
3 < x < 7

This is the general approach which can be used when you see multiplications in inequalities.

Regards,

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Re: x2 - 4x + 3 < 0 [#permalink]

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30 May 2010, 04:27
5
KUDOS
Bunuel wrote:
dimitri92 wrote:
If x is an integer, what is the value of x?

(1) x2 - 4x + 3 < 0
(2) x2 + 4x +3 > 0

Given: $$x=integer$$.

(1) $$x^2-4x+3<0$$ --> $$1<x<3$$--> as $$x$$ is an integer then $$x=2$$. Sufficient.

(2) $$x^2+4x+3>0$$ --> $$x<-3$$ or $$x>-1$$ --> multiple values are possible for integer $$x$$. Not sufficient.

With reference to your statement $$1<x<3$$ above, I calculated the range as $$x<1$$ or $$x<3$$
What I did wrong?
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Re: If x is an integer, what is the value of x? (1) x^2 - 4x + 3 [#permalink]

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03 Aug 2012, 23:52
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bhavinshah5685 wrote:
Hi, Can u please eloaborate above highlighted part???. I m still confused how did u rearranged 9)(8)(7)(3) > 0 in to number line and subsequent results?? Help appriciated..!!!

Plug x = 10 for the expression:

(x-1)(x-2)(x-3)(x-7) < 0
= (10-1)(10-2)(10-3)(10-7)
You get 9*8*7*3 which means the expression is positive when x =10
The roots of the expression (x-1)(x-2)(x-3)(x-7) < 0 are: 1,2,3 and 7
Since 10 >7, the expression is positive when the value of x is greater than 7.
From then on, just flip the sign every time you hit a root.
So:
from 3 to 7 the expression is -ive
from 2 to 3 the expression is +ive etc..
The idea is to start from the root with the highest absolute value, find out what the sign for the expression is and to work from there..

For more on this check out cyberjadugar's signature: Solving Inequalities

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Re: What is the value of x? [#permalink]

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30 May 2010, 04:06
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Hussain15 wrote:
If $$x$$ is an integer, what is the value of $$x$$ ?

(1) $$x^2 - 4x + 3 < 0$$
(2) $$x^2 + 4x +3 > 0$$

IMO A

1st equation gives (x-1)*(x-3) <0 which is only possible when x=2

2nd gives (x+1)*(x+3) > 0 which is possible when x< -3 and x> -1 thus multiple values.

Thus A

OA pls.
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Re: If x is an integer, what is the value of x? (1) x^2 - 4x + 3 [#permalink]

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07 Jul 2013, 22:20
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(1) reduces to (x-3)(x-1) < 0 which is possible only when x-3 and x-1 are of the opposite signs which => 1<x<3. Since x is an integer, we have a solution for x=2.

(2) reduces to (x+3)(x+1) > 0 which is possible for x < -3 or x > -1 which has many solutions, Not sufficient.

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Re: If x is an integer, what is the value of x? (1) x^2 - 4x + 3 [#permalink]

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31 Aug 2015, 03:49
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Expert's post
Product of 2 terms <0 , then the variable lies between the given points.

(x-a) (x-b)<0, (considering a<b, in the number line)

1st Condition

Either x-a<0 i.e x<a or x-b>0 i.e x>b - NO common region

2nd Condition

Either x-a>0 i.e x>a or x-b<0 i.e x<b - a<x<b

Hope this helps.
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Re: x2 - 4x + 3 < 0 [#permalink]

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23 May 2010, 01:07
C

St. 1: insuff --> x< 3 or x < 1

St. 2: insuff --> x> -3 or x > -1

St. 1 & 2 : Sufficient --> -1 < x < 1 and as x is an integer, x = 0.

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Re: x2 - 4x + 3 < 0 [#permalink]

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30 May 2010, 23:01
Great explanation Bunuel!! Kudos!
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Re: x2 - 4x + 3 < 0 [#permalink]

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31 May 2010, 04:38
Thanks for the explanation.

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Re: x2 - 4x + 3 < 0 [#permalink]

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15 Mar 2011, 20:04
From (1) we have (x-3)(x-1) < 0

=> 3 < x < 1, and there is only one integer between 1 and 3 which is 2, so (1) is sufficient.

From (2) we have (x+1)(x+3) > 0 So either x > -1 or x < - 3, which is not enough

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Re: x2 - 4x + 3 < 0 [#permalink]

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25 Jun 2011, 00:04
Kudos Bunuel. I struggled with these questions always. But now its clear.
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Re: x2 - 4x + 3 < 0 [#permalink]

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20 Apr 2012, 10:15
You are a genius!!! I have been struggling with inequalities particularly quadratics, this solved my issue!!

Bunuel wrote:
Hussain15 wrote:
Bunuel wrote:

How to solve quadratic inequalities - Graphic approach.

$$x^2-4x+3<0$$ is the graph of parabola and it look likes this:
Attachment:
en.plot (1).png

Intersection points are the roots of the equation $$x^2-4x+3=0$$, which are $$x_1=1$$ and $$x_2=3$$. "<" sign means in which range of $$x$$ the graph is below x-axis. Answer is $$1<x<3$$ (between the roots).

If the sign were ">": $$x^2-4x+3>0$$. First find the roots ($$x_1=1$$ and $$x_2=3$$). ">" sign means in which range of $$x$$ the graph is above x-axis. Answer is $$x<1$$ and $$x>3$$ (to the left of the smaller root and to the right of the bigger root).

This approach works for any quadratic inequality. For example: $$-x^2-x+12>0$$, first rewrite this as $$x^2+x-12<0$$ (so that the coefficient of x^2 to be positive. It's possible to solve without rewriting, but easier to master one specific pattern).

$$x^2+x-12<0$$. Roots are $$x_1=-4$$ and $$x_1=3$$ --> below ("<") the x-axis is the range for $$-4<x<3$$ (between the roots).

Again if it were $$x^2+x-12>0$$, then the answer would be $$x<-4$$ and $$x>3$$ (to the left of the smaller root and to the right of the bigger root).

Hope it helps.

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Re: If x is an integer, what is the value of x? (1) x^2 - 4x + 3 [#permalink]

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23 Apr 2012, 14:10
Amazing explanation.... wow.

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Re: x2 - 4x + 3 < 0 [#permalink]

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12 May 2012, 11:09
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kartik222 wrote:
@Bunuel,
I don't understand how you calculated the roots this equation: $$x^2+x-12<0$$. your answer: Roots are $$x_1=-4$$ and $$x_1=3$$

$$x^2+x-12<0$$
(x-4)(x+3)<0
so, two point now will be 4 and -3

Now the we can plot the graph and get to the answer.

Thanks!

Hope it helps.
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Re: If x is an integer, what is the value of x? (1) x^2 - 4x + 3 [#permalink]

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09 Jun 2012, 03:24
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Hi,

I won't mind drawing the parabola for a quadratic equation. But a better approach is to check the sign of expression on a number line.

for example: (x-1)(x-2)(x-3)(x-7) < 0

To check the intervals in which this inequality holds true, we need to pick only one value from the number line.
Lets say x = 10, then (9)(8)(7)(3) > 0, in every alternate interval the sign would be + for the above expression

---(+)-----1--(-)--2---(+)--3----(-)-------7----(+)------

Thus, inequality would hold true in the intervals:
1 < x < 2
3 < x < 7

This is the general approach which can be used when you see multiplications in inequalities.

Regards,

Nice little trick! And useful too! This combined with Bunuel's graphical approach to quadratic inequalities make these problem types so easy now..

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Re: If x is an integer, what is the value of x? (1) x^2 - 4x + 3 [#permalink]

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03 Aug 2012, 18:59
Wow. I've been struggling for hours on this.

This is the best explanation I've seen of any GMAT related stuff ever.

So cool!!!!!!!!!! Thanks

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Re: If x is an integer, what is the value of x? (1) x^2 - 4x + 3 [#permalink]

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03 Aug 2012, 23:19
Hi,

I won't mind drawing the parabola for a quadratic equation. But a better approach is to check the sign of expression on a number line.

for example: (x-1)(x-2)(x-3)(x-7) < 0

To check the intervals in which this inequality holds true, we need to pick only one value from the number line.
Lets say x = 10, then (9)(8)(7)(3) > 0, in every alternate interval the sign would be + for the above expression

---(+)-----1--(-)--2---(+)--3----(-)-------7----(+)------

Thus, inequality would hold true in the intervals:
1 < x < 2
3 < x < 7

This is the general approach which can be used when you see multiplications in inequalities.

Regards,

Hi, Can u please eloaborate above highlighted part???. I m still confused how did u rearranged 9)(8)(7)(3) > 0 in to number line and subsequent results?? Help appriciated..!!!

Kudos [?]: 132 [0], given: 21

Re: If x is an integer, what is the value of x? (1) x^2 - 4x + 3   [#permalink] 03 Aug 2012, 23:19

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