Is it Right!! Inequalities

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BTG14: Senior | Next Rank: 100 Posts; Posts: 45; Joined: Fri Apr 06, 2012 5:28 amFollowed by: 2 members; Thanked: 3 times; GMAT Score:500

by BTG14 » Sun Feb 03, 2013 7:49 pm

Hi,
When i was going through Inequalities chapter I found one note.

If |x| > a then x>a or x<-a. Can we modify it has
If |x| > a then x>a or x<a. Is it right.

Please let me know if it is wrong.

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Source: Beat The GMAT — Problem Solving | Sun Feb 03, 2013 7:49 pm

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sun Feb 03, 2013 11:08 pm

BTG14 wrote:Hi,
When i was going through Inequalities chapter I found one note.

If |x| > a then x>a or x<-a. Can we modify it has
If |x| > a then x>a or x<a. Is it right.

Please let me know if it is wrong.

The part in blue is true only if a is positive.
If a is negative then we cannot conclude that x < -a based on the fact that |x| > a.
For example, it's true that |2| > -1, but it's not true that 2 < -(-1)

Similarly, there's a problem with the part in green.
If |x| > a, we cannot conclude that x<a
For example, it's true that |2| > 1, but we cannot conclude that 2 < 1

Here's what you need to know: (note: the rules apply only if a > 0):
1) If |x| < a, then -a < x < a
2) If |x| > a, then x > a or x < -a

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Mon Feb 04, 2013 4:51 pm

When you're thinking about absolute value (particularly with inequalities), it's helpful to think in terms of a number line. For example, |x| > 2 would look like this:

<--------|--|--|--------->

If x is positive, it's greater than positive 2. If x is negative, it's less than -2.

You also want to think conceptually about what those numbers on a number line would mean. You asked:

If |x| > a then x>a or x<-a. Can we modify it has
If |x| > a then x>a or x<a. Is it right.

But how could x be greater than a and less than a? This wouldn't make sense - but it's a very common mistake to make algebraically. Make sure that you're not just thinking algebraically, but also asking - do those numbers make sense?

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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