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madhan_dc: Junior | Next Rank: 30 Posts; Posts: 24; Joined: Mon Jan 24, 2011 1:37 pm

Algebra Q

by madhan_dc » Sun Jan 30, 2011 11:02 am

Solve for x

1. 6> x+4 > 4

2. 2x > x+10 > -x

answer is 1. 0<x<2
2. x>10

Can someone explain to me how?

the way i went about doing this is

1. 6 > x+4 -4 > 0

therefore 6 > x >0

2. 2x > x+10 > -x

2x > x + x+ 10 > 0

2x > 2x + 10 > 0

and i didn't know where to go from here. Clearly i am missing something. Please help. Thanks.

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Source: Beat The GMAT — Problem Solving | Sun Jan 30, 2011 11:02 am

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Sun Jan 30, 2011 11:17 am

madhan_dc wrote:Solve for x

1. 6> x+4 > 4
2. 2x > x+10 > -x

Note that each inequality is composed of two inequalities.

1. 6 > (x + 4) > 4
=> (6 - 4) > (x + 4 - 4) > (4 - 4) ................ 4 is subtracted from each term
=> 2 > x > 0

In this case there were two inequalities, 6 > (x + 4) and (x + 4) > 4.

2. 2x > (x + 10) > -x
=> (2x - x) > (x + 10 - x) > -x - x
=> x > 10 > -2x
=> x > 10

In this case also there were two inequalities, 2x > (x + 10) and (x + 10) > -x
From the result x > 10 > -2x, we only took x > 10 as this ensures the other, i.e. -2x will be always less than 10 if x is greater than 10.

Hope it is clear now.

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

by Stuart@KaplanGMAT » Sun Jan 30, 2011 11:32 am

madhan_dc wrote:Solve for x

1. 6> x+4 > 4

2. 2x > x+10 > -x

Hi,

when dealing with 3 part inequalities, here's the key rule to remember: whatever you do to one part, you must do to all 3 parts.

Since we're solving for x, we want to isolate x, i.e. strip the part of the inequality that contains x of everything except x itself.

1. 6> x+4 > 4

Here, to isolate x, we want to get rid of the "+4". To eliminate +4, we need to subtract 4 from that term. What we do to 1 term we must do to all 3 terms, so:

6 - 4 > x + 4 - 4 > 4 - 4

and

2 > x > 0

done!

2. 2x > x+10 > -x

This one is a bit more complicated, since there's no way to collect all of the Xs in the same part of the inequality. We can begin to simplify by adding x to each term:

2x + x > x + 10 + x > -x + x

3x > 2x + 10 > 0

Now, because we know that 3x > 0 (we can ignore the middle piece since the arrows all point in the same direction), we know that x > 0, i.e. x is positive. Since x is positive, 2x + 10 is greater than 0. Because the 2nd term is greater than the 3rd term, we can actually ignore that part of the inequality and just solve for:

3x > 2x + 10

Subtract 2x from both terms:

3x - 2x > 2x + 10 - 2x

x > 10

done!