Absolute Values and Equalities

[mindful]

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?

[GMATGuruNY]

Check my post here:

https://www.beatthegmat.com/algebra-hard ... 25445.html

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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

[mindful]

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...!

[theCEO]

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

[mindful]

Thank you, CEO. This is lovely!!!