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# Inequalities and Roots

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Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 5468
Location: Pune, India
Followers: 1339

Kudos [?]: 6803 [1] , given: 177

Re: Inequalities and Roots [#permalink]  25 Jul 2013, 20:26
1
KUDOS
Expert's post
gmatter0913 wrote:
Hi Karishma,

I tried the problem x-1 < sqrt (7-x) as below:

As 7-x is under sqrt, it is +ve. Therefore, 7-x>=0 ; x<=7 ----->(1)

x-1 can be -ve or +ve

When x-1<=0; x<=1 --------> (2)

When x-1>=0; x>=1 --------> (3)

As both sides are +ve, we can square both the sides

(x-1)^2 < 7-x
x^2 -x -6<0
(x-3)(x+2)<0

-2<x<3 ------------>(4)

The answer to this problem is (x<3). I am not sure how to arrive at that from hereon. Could you please help me?

You have done the process correctly. Now you need to understand what this implies.

You got x <= 7

Case 1: x-1< 0
When x-1< 0; x < 1
Note that when x-1 is negative, it will always be less than $$\sqrt{(7-x)}$$
So whenever x<1, the inequality will always hold.

Case 2: x-1 >= 0
If x-1 is non negative, we can square the inequality.
From this, you get -2<x<3.
The inequality holds in this range.

From the two cases, we see that the inequality holds for the range x < 3.
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Senior Manager
Joined: 12 Mar 2010
Posts: 385
Concentration: Marketing, Entrepreneurship
GMAT 1: 680 Q49 V34
Followers: 2

Kudos [?]: 65 [0], given: 87

Re: Inequalities and Roots [#permalink]  26 Jul 2013, 01:05
Hi Karishma,

I have one more doubt on my solution posted earlier.

Quote:
I tried the problem x-1 < sqrt (7-x) as below:

As 7-x is under sqrt, it is +ve. Therefore, 7-x>=0 ; x<=7 ----->(1)

x-1 can be -ve or +ve

When x-1<=0; x<=1 --------> (2)

When x-1>=0; x>=1 --------> (3)

As both sides are +ve, we can square both the sides

(x-1)^2 < 7-x
x^2 -x -6<0
(x-3)(x+2)<0

-2<x<3 ------------>(4) Shouldn't this be 1<=x<3 (as x>=1 is the pre-supposed condition to square them)
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 5468
Location: Pune, India
Followers: 1339

Kudos [?]: 6803 [1] , given: 177

Re: Inequalities and Roots [#permalink]  26 Jul 2013, 03:23
1
KUDOS
Expert's post
gmatter0913 wrote:
Hi Karishma,

I have one more doubt on my solution posted earlier.

Quote:
I tried the problem x-1 < sqrt (7-x) as below:

As 7-x is under sqrt, it is +ve. Therefore, 7-x>=0 ; x<=7 ----->(1)

x-1 can be -ve or +ve

When x-1<=0; x<=1 --------> (2)

When x-1>=0; x>=1 --------> (3)

As both sides are +ve, we can square both the sides

(x-1)^2 < 7-x
x^2 -x -6<0
(x-3)(x+2)<0

-2<x<3 ------------>(4) Shouldn't this be 1<=x<3 (as x>=1 is the pre-supposed condition to square them)

Most certainly. The only reason I don't care about the values from -2 to 1 is that these values are already covered in the first case. We know they already hold for the inequality. We only get 1 to 3 extra values and that's what we care about.

In a stand alone question, the presupposed condition must be satisfied (x <= 7 AND x >= 1)
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Veritas Prep GMAT course is coming to India. Enroll in our weeklong Immersion Course that starts March 29!

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GMAT Club Legend
Joined: 09 Sep 2013
Posts: 4791
Followers: 296

Kudos [?]: 52 [0], given: 0

Re: Inequalities and Roots [#permalink]  07 Aug 2014, 05:47
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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Inequalities and Roots   [#permalink] 07 Aug 2014, 05:47

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