Doubt in Inequality

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seema19: Junior | Next Rank: 30 Posts; Posts: 23; Joined: Wed Sep 14, 2011 12:34 am

by seema19 » Sat Nov 19, 2011 10:32 am

1. If x^2 > 1, then what are the limits of that inequality?
2. If (5X - 2)(3X + 1) > 0, then what are the limits of that inequality?

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Source: Beat The GMAT — Data Sufficiency | Sat Nov 19, 2011 10:32 am

rijul007: Legendary Member; Posts: 588; Joined: Sun Oct 16, 2011 9:42 amFollowed by: 9 members; Location: New Delhi, India; Thanked: 130 times; GMAT Score:720

by rijul007 » Sat Nov 19, 2011 10:42 am

1. If x^2 > 1, then what are the limits of that inequality?

x^2 > 1
x^2 - 1 > 0
(x+1)(x-1) > 0

Take cases

Case 1: x > 1
(x+1)(x-1) will be positive

Case 2 -1<x<1
(x+1)(x-1) will be ngative

Case 3 x<-1
(x+1)(x-1) will be positive

Hence
x^2 > 1 for x<-1 and x>1

2. If (5X - 2)(3X + 1) > 0, then what are the limits of that inequality?
similarly take cases in these too..

you'll get the ans
x<-1/2 and x>2/5

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kanwar86: Senior | Next Rank: 100 Posts; Posts: 99; Joined: Thu Jun 09, 2011 2:09 pm; Thanked: 11 times

by kanwar86 » Sat Nov 19, 2011 10:46 am

IMO
1) x<-1 or x>1
2) x<-1/3 or x>2/5

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zooki: Junior | Next Rank: 30 Posts; Posts: 29; Joined: Fri Oct 14, 2011 10:15 amFollowed by: 1 member; Thanked: 4 times

by zooki » Sat Nov 19, 2011 11:14 am

for Q 2: x=-1/3 does not satisfy the inequality.
neither does x=2/5

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vishal.pathak: Master | Next Rank: 500 Posts; Posts: 176; Joined: Thu Sep 22, 2011 5:32 am; Thanked: 5 times

by vishal.pathak » Sat Nov 19, 2011 11:36 am

seema19 wrote:1. If x^2 > 1, then what are the limits of that inequality?
2. If (5X - 2)(3X + 1) > 0, then what are the limits of that inequality?

1.if x^2 > 1 then -infinity < x < -1 and 1 < x < infinity
2. if (5x - 2) (3x + 1) > 0
then
either both (5x-2) and (3x+1) have to be +ve or both have to be -ve for the final product to b +ve (>0)
when both +ve x > 2/5 (5x-2 > 0), x>-1/3(3x+1 > 0)
so we see that x > 2/5 satisfies both x > 2/5 and x>-1/3
when both -ve x < 2/5, x < -1/3
so we see that x < -1/3 satisfies both x<2/5 and x<-1/3
so the range is x < -1/3 , x > 2/5

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seema19: Junior | Next Rank: 30 Posts; Posts: 23; Joined: Wed Sep 14, 2011 12:34 am

by seema19 » Sat Nov 19, 2011 11:42 am

X^2 > 1
X^2 - 1 > 0
(X + 1)(X - 1) > 0
Thus, X > -1 and X > +1.
However, according to the answer,this is wrong. Can someone plz explain why this is wrong.

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kanwar86: Senior | Next Rank: 100 Posts; Posts: 99; Joined: Thu Jun 09, 2011 2:09 pm; Thanked: 11 times

by kanwar86 » Sat Nov 19, 2011 11:57 am

seema19 wrote:X^2 > 1
X^2 - 1 > 0
(X + 1)(X - 1) > 0
Thus, X > -1 and X > +1.
However, according to the answer,this is wrong. Can someone plz explain why this is wrong.

For expression (X-1)(X+1) to be greater than zero either X should be >1 or <-1
In your case for 1>X>-1, the term (X+1)>0 but (X-1)<0, which implies the product, X^2-1, is less than zero. Hence, -1<x<1 can't be the solution.

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vishal.pathak: Master | Next Rank: 500 Posts; Posts: 176; Joined: Thu Sep 22, 2011 5:32 am; Thanked: 5 times

by vishal.pathak » Sat Nov 19, 2011 12:06 pm

seema19 wrote:X^2 > 1
X^2 - 1 > 0
(X + 1)(X - 1) > 0
Thus, X > -1 and X > +1.
However, according to the answer,this is wrong. Can someone plz explain why this is wrong.

for (X + 1)(X - 1) > 0 either both ((X + 1) & (X - 1)) have to be +ve or both have to be -ve so that the final product is +ve
+ve*+ve = +ve
-ve*-ve = -ve
if both are +ve then x+1>0 and x-1>0
or x > -1 and x>1
So u can see that x>1 satisfies both x > -1 and x>1 (so range from this is x>1, lets call it eqn (1))
Now if both ((X + 1) & (X - 1)) are -ve
then x<-1 and x<1
So u can see that x<-1 satisfies both x<-1 and x<1 (so range from this is x<-1, lets call it eqn (2))
From eqn (1) and eqn(2)
x<-1 and x >1

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pemdas: Legendary Member; Posts: 1084; Joined: Fri Apr 15, 2011 2:33 pmFollowed by: 21 members; Thanked: 158 times

by pemdas » Sat Nov 19, 2011 12:09 pm

likewise |x|>|1| you solve by squaring both sides you can consider the roots here
for |x|>|1| the only solutions are mode must be greater than the absolute value of 1 regardless of signs, hence x<-1 and x>1

seema19 wrote:X^2 > 1
X^2 - 1 > 0
(X + 1)(X - 1) > 0
Thus, X > -1 and X > +1.
However, according to the answer,this is wrong. Can someone plz explain why this is wrong.

Success doesn't come overnight!

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kanwar86: Senior | Next Rank: 100 Posts; Posts: 99; Joined: Thu Jun 09, 2011 2:09 pm; Thanked: 11 times

by kanwar86 » Sat Nov 19, 2011 12:20 pm

Please note that the solution to 1)is X<-1 OR X>1 instead of X<-1 AND X>1 (which is not possible as both sets are mutually exclusive)

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun Nov 20, 2011 5:32 am

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