Absolute Values and Equalities

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mindful: Senior | Next Rank: 100 Posts; Posts: 52; Joined: Wed Jun 24, 2015 10:57 pmFollowed by: 1 member; Thanked: 3 times

by mindful » Tue Sep 29, 2015 11:41 pm

Here is a simple question. Would like to see someone share the process of arriving at an answer. Thx.

When is |x-5| equal to 5-x?

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Source: Beat The GMAT — Problem Solving | Tue Sep 29, 2015 11:41 pm

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 » Wed Sep 30, 2015 3:13 am

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by [email protected] » Wed Sep 30, 2015 11:00 am

Hi mindful,

Sometimes the easiest way to deal with a complex-looking situation is to just 'play around' with it a bit by TESTing VALUES. You might be surprised how quickly you can deduce a pattern if you just come up with a few examples.

You're essentially asking what values of X will fit the following equation:

|X-5| = 5-X

I'm going to ask you a few questions; see how long it takes you to answer them?

1) What do you 'know' about Absolute Values? What 'end results' are possible? What 'end results' are NOT?
2) What happens when X = 5?
3) What happens when X = 6 or 7 or 8?
4) What happens when X is LESS than 5? Come up with 3 examples (one positive, one 0 and one negative).

5) After working through the above examples, what deductions have you made about X?

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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mindful: Senior | Next Rank: 100 Posts; Posts: 52; Joined: Wed Jun 24, 2015 10:57 pmFollowed by: 1 member; Thanked: 3 times

by mindful » Tue Oct 06, 2015 3:57 am

Thank you. That's a good suggestion about taking one positive, one negative, and 0 as a number. When I took -8 as a choice, I got positive 13 on both sides.

What bothers me is that the first time I attempted it algebraically. Why didn't I get an answer using that method.

I said: |x-5| = 5 - x

There are two possibilities:

x-5 = 5 - x
so 2x = 10
x = 5

OR

x-5 = x-5

x-x = -5+5

and I didn't really know what to do next.

The answer I got then was that both sides are equal if x = 5. But as we see the real answer is whenever x< or equal to 5, we get |x-5| as 5-x.

How can I do this algebraically?

Thanks...!

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theCEO: Master | Next Rank: 500 Posts; Posts: 363; Joined: Sun Oct 17, 2010 3:24 pmFollowed by: 3 members; Thanked: 115 times

by theCEO » Tue Oct 06, 2015 3:32 pm

mindful wrote:Thank you. That's a good suggestion about taking one positive, one negative, and 0 as a number. When I took -8 as a choice, I got positive 13 on both sides.

What bothers me is that the first time I attempted it algebraically. Why didn't I get an answer using that method.

I said: |x-5| = 5 - x

There are two possibilities:

x-5 = 5 - x
so 2x = 10
x = 5

OR

x-5 = x-5

x-x = -5+5

and I didn't really know what to do next.

The answer I got then was that both sides are equal if x = 5. But as we see the real answer is whenever x< or equal to 5, we get |x-5| as 5-x.

How can I do this algebraically?

Thanks...!

algebraically
|x-5|= 5-x
|x-5|>= 0, therefore 5-x >= 0

when is 5-x >= 0?
5-x >= 0
-x >= -5
x <= 5

Thefore |x-5| equal to 5-x when x<=5

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mindful: Senior | Next Rank: 100 Posts; Posts: 52; Joined: Wed Jun 24, 2015 10:57 pmFollowed by: 1 member; Thanked: 3 times

by mindful » Tue Oct 06, 2015 8:29 pm

Thank you, CEO. This is lovely!!!

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