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Inequalities - solving for x

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rganesh: Newbie | Next Rank: 10 Posts; Posts: 3; Joined: Sun Jan 06, 2008 7:58 pm

by rganesh » Tue Apr 22, 2008 4:44 am

Hi there,
I keep looking at this problem and i know it should be pretty simple but I can't seem to wrap my head around solving for x.

Two questions:

1) 6>x+4>4

2) 2x>x+10>-x

Thank you!

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Source: Beat The GMAT — Problem Solving | Tue Apr 22, 2008 4:44 am

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

Re: Inequalities - solving for x

by Stuart@KaplanGMAT » Tue Apr 22, 2008 8:45 am

rganesh wrote:Hi there,
I keep looking at this problem and i know it should be pretty simple but I can't seem to wrap my head around solving for x.

Two questions:

1) 6>x+4>4

2) 2x>x+10>-x

Thank you!

We treat 3 part inequalities exactly as we do 2 part inequalities: whatever we do to 1 part, we must do to all 3 parts. Further, if we multiply or divide by a negative, we reverse the inequalities.

Assuming that our task is to isolate x:

1) 6 > x+4 > 4

let's subtract 4 from each part:

6-4 > x+4-4 > 4-4
2 > x > 0

2) 2x>x+10>-x

This one is trickier. Neither adding nor subtracting x from every part will leave us with x in only 1 part.

If we add x to each part, we get:

2x + x > x+10 +x > -x + x
3x > 2x + 10 > 0

If we subtract x from each part, we get:

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

x > 10 > -2x

We may be tempted to divide each part by x, but we have to be very careful when variables are involved, since we don't know if x is positive, negative or 0.

Is there an actual question that goes along with this one? If we knew what kind of answer we were looking for, it would be easier to figure out what to do.

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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AleksandrM: Legendary Member; Posts: 566; Joined: Fri Jan 04, 2008 11:01 am; Location: Philadelphia; Thanked: 31 times; GMAT Score:640

by AleksandrM » Tue Apr 22, 2008 9:10 am

I got x > 5 for the second one. Did anyone else?

https://second-lap.blocked/

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

by Stuart@KaplanGMAT » Tue Apr 22, 2008 9:41 am

AleksandrM wrote:I got x > 5 for the second one. Did anyone else?

If you plug in 6, you get:

12 > 16 > -6

So, x > 5 isn't the correct interpretation.

In retrospect, applying a bit of logic will make our lives easier.

The third part is actually redundant.

If we look at the first two parts, we get:

2x > x + 10

2x - x > x + 10 -x

x > 10

Incorporating the 3rd part would have let us determine that x > 0, but since x must also be greater than 10, x > 0 gets subsumed and we can ignore it.

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