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by [email protected] » Wed Jul 30, 2014 7:57 pm
Hi kamalakarthi,

This question combines some basic algebra skills with some Number Properties. Since we're dealing with inequalities (instead of equations), chances are that we'll have to do a little more work than normal to answer this question.

We're told that X is an INTEGER. We're asked for the value of X.

Fact 1: X^2 - 4X + 3 < 0

You can factor this quadratic into 2 pieces:

(X-1)(X-3) < 0

Now comes the Number Property....

For the above product to be LESS than 0, ONE of the pieces must be POSTIVE and the OTHER must be NEGATIVE. That's the ONLY way to get a result that is less than 0. So let's talk about the possibilities....remember that X must be an INTEGER

X can't be 3, 4, 5, 6, etc. because then you'll either have two positives or a 0.
X CAN be 2, since (1)(-1) is less than 0
X can't be 1, because then you'd have a 0.
X can't be 0, -1, -2, -3, -4 etc. because then you'd have two negatives.
The only answer is 2.
Fact 1 is SUFFICIENT.

Fact 2: X^2 + 4X + 3 > 0

From here, we don't have to do nearly as much work. X could be ANY POSITIVE INTEGER, so there are an infinite number of answers.
Fact 2 is INSUFFICIENT

Final Answer: A

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by Brent@GMATPrepNow » Wed Jul 30, 2014 8:03 pm
kamalakarthi wrote:If x is an integer, what is the value of x?

(1) x² - 4x + 3 < 0

(2) x² + 4x +3 > 0

Target question: What is the value of x?

Given: x is an integer

Statement 1: x² - 4x + 3 < 0
First solve the EQUATION x² - 4x + 3 = 0
(x - 1)(x - 3) = 0
So, x = 1 and x = 3 satisfy the EQUATION x² - 4x + 3 = 0

Now check the RANGES OF VALUES when x equals neither 1 nor 3
case a: x < 1
Try x = 0
If x = 0, then (x - 1)(x - 3) is POSITIVE. This means that for ANY value of x where x < 1, (x - 1)(x - 3) will always be POSITIVE

case b: 1 < x < 3
Try x = 2
If x = 2, then (x - 1)(x - 3) is NEGATIVE. This means that for ANY value of x where 1 < x < 3, (x - 1)(x - 3) will always be NEGATIVE

case c: 3 < x
Try x = 4
If x = 4, then (x - 1)(x - 3) is POSITIVE. This means that for ANY value of x where 3 < x, (x - 1)(x - 3) will always be POSITIVE

Statement 1 tells us that x² - 4x + 3 [aka (x - 1)(x - 3)] is NEGATIVE (case b).
This means that 1 < x < 3
Since we're told that x is an integer, it must be the case that x = 2
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x² + 4x +3 > 0
First solve the EQUATION x² + 4x + 3 = 0
(x + 1)(x + 3) = 0
So, x = -1 and x = -3 satisfy the EQUATION x² + 4x + 3 = 0

Now check the RANGES OF VALUES when x equals neither -1 nor -3
case a: x < -3
Try x = -4
If x = -4, then (x + 1)(x + 3) is POSITIVE. This means that for ANY value of x where x < -3, (x + 1)(x + 3) will always be POSITIVE

case b: -3 < x < -1
Try x = -2
If x = -2, then (x + 1)(x + 3) is NEGATIVE. This means that for ANY value of x where -3 < x < -1, (x + 1)(x + 3) will always be NEGATIVE

case c: -1 < x
Try x = 0
If x = 0, then (x + 1)(x + 3) is POSITIVE. This means that for ANY value of x where -1 < x, (x + 1)(x + 3) will always be POSITIVE

Statement 2 tells us that x² + 4x + 3 [aka (x + 1)(x + 3)] is POSITIVE (cases a & c).
This means that EITHER x < -3 OR x > -1
This means that there are several different possible values of x.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent
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by GMATinsight » Wed Jul 30, 2014 9:31 pm
kamalakarthi wrote:If x is an integer, what is the value of x ?

(1) x² - 4x + 3 < 0

(2) x² + 4x +3 > 0

Can u please help me solving this.
Question : x = ?

Statement 1) x² - 4x + 3 < 0
i.e. (x-1) (x-3) < 0
i.e. Product of two numbers (x-1) and (x-3) is negative for all Integer values of x
Which is possible only if one of (x-1) and (x-3) is positive and other is Negative

i.e. EITHER (x-1) < 0 as well as (x-3) > 0
i.e. Either (x) < 1 as well as (x) > 3
i.e. Which is Impossible and therefore inconsistent

OR (x-1) > 0 as well as (x-3) < 0
i.e. Either (x) > 1 as well as (x) < 3
i.e. x = 2
SUFFICIENT


Statement 2) x² + 4x +3 > 0
i.e. (x + 1)(x + 3) > 0
i.e. Product of two numbers (x-1) and (x-3) is Positive for all Integer values of x
Which is possible only if both (x-1) and (x-3) are positive or both are Negative
i.e. EITHER (x + 1) > 0 as well as (x + 3) > 0
i.e. (x) > -1 as well as (x) > -3
INFINITE POSSIBILITIES

i.e. OR (x + 1) < 0 as well as (x + 3) < 0
i.e. (x) < -1 as well as (x) < -3
INFINITE POSSIBILITIES
INSUFFICIENT

Answer: Option A
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by perwinsharma » Wed Jul 30, 2014 11:18 pm
If x is an integer, what is the value of x ?

(1) x2 - 4x + 3 < 0

(2) x2 + 4x +3 > 0
Considering statement (1) alone:

x2 - 4x + 3 < 0
=> (x - 1) (x - 3) < 0
For the product of two numbers to be negative, exactly 1 has to be negative and the other has to be positive.

Case 1:
If x - 1 < 0 and x - 3 > 0
=> x < 1 and x > 3 (Not possible; no integer is allowed in such a range)

Case 2:
If x - 1 > 0 and x - 3 < 0
=> x > 1 and x < 3
=> x = 2 (The only integer allowed in this rage)

SUFFICIENT
BCE goes out

Considering statement (2) alone:
x2 + 4x +3 > 0
=> (x + 1) (x + 3) > 0

For the product of two integers to be positive, either both the numbers should be positive or both the numbers should be negative.

Case 1:
If x + 1 > 0 and x + 3 > 0
=> x > -1 and x > -3
=> x = 0, 1, 2, 3, 4,.....(multiple posible values)

Case 2:
If x + 1 < 0 and x + 3 < 0
=> x < -1 and x < -3
=> x = -4, -5, -6, .......(multiple possible values)

INSUFFICIENT

The answer is (A).
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by GMATGuruNY » Thu Jul 31, 2014 3:30 am
kamalakarthi wrote:If x is an integer, what is the value of x ?

(1) x2 - 4x + 3 < 0

(2) x2 + 4x +3 > 0
Alternate approach:

Statement 1: x² - 4x + 3 < 0
Thus:
x² < 4x - 3.

Since the value of x² cannot be less than 0, 4x - 3 must be POSITIVE:
4x - 3 > 0
x > 3/4.
Implication:
The value of x is constrained to POSITIVE INTEGERS.

Plugging x=1 into x² < 4x - 3, we get 1 < 1.
Plugging x=2 into x² < 4x - 3, we get 4 < 5.
Plugging x=3 into x² < 4x - 3, we get 9 < 9.
Plugging x=4 into x² < 4x - 3, we get 16 < 13.
Plugging x=5 into x² < 4x - 3, we get 25 < 17.
At this point, we can see that only the option in red is viable.
Thus, x=2.
SUFFICIENT.

Statement 2: x² + 4x + 3 > 0
If x=0, then x² + 4x + 3 = 3, which is greater than 0.
If x=1, then x² + 4x + 3 = 8, which is greater than 0.

Since x can be equal to different values, INSUFFICIENT.

The correct answer is A.

These sorts of problems can also be solved by determining the CRITICAL POINTS.
Examples:
https://www.beatthegmat.com/inequality-c ... 89518.html
https://www.beatthegmat.com/knewton-q-t89317.html
https://www.beatthegmat.com/which-is-true-t89111.html
https://www.beatthegmat.com/inequalities ... 76123.html
https://www.beatthegmat.com/if-x-is-not- ... 65585.html
https://www.beatthegmat.com/inequality-q ... 74830.html
https://www.beatthegmat.com/if-x-is-posi ... 70254.html
https://www.beatthegmat.com/inequalities-t115175.html (2nd post)
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