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Confused about rules for absolute values - help!

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Confused about rules for absolute values - help!

by suzeemunkee » Sun Jul 11, 2010 4:22 pm
Hi guys -

I'm having trouble understanding the rules around solving absolute value questions. Can someone help me?

OK, so first consider this question from MGMAT:

Is x > 0?

(1) |x + 3| = 4x - 3

(2) |x + 1| = 2x - 1

The answer is D; each statement alone is sufficient. The answer explanation says that even though each statement gives two solutions (statement 1 gives x = 2 and 0; statement 2 gives x = 3 and -1), we need to verify that both solutions are valid by plugging in each solution into the original equation of each statement. When we do this, we see that x = 2 is the only valid solution for each statement.

With that in mind, now consider a similar question from Smart GMAT:

What is the value of x?

(1) |6 - 3x| = x - 2

(2) |5x + 3| = 2x + 9

This time, the answer is only A.

Why isn't the answer D? Even though Statement 2 gives two solutions (x = 2 and -12/7), can't I verify the solutions by plugging into the original equation? When I do that, I find that only x = 2 is valid.

Can someone explain this discrepancy to me? Thanks all!
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by barcebal » Sun Jul 11, 2010 5:31 pm
Let's take it from the top. Remember we must answer each statement individually.

FIRST QUESTION

Statement (1) We get solutions of x=2 and x=0. We plug both back in and find that x=0 is actually not a valid solution which means x must equal 2. This answers the question is x>0. Statement 1 is sufficient.

Statement (2) We get solutions of x=2 and x=0. Same thing happens. We plug both back in and find that x=0 is actually not a valid solution which means x must equal 2. This answers the question is x>0. Statement 1 is sufficient.

Both statements, alone, are sufficient. Answer D

SECOND QUESTION (I think this is where you are getting confused.)

Statement (1) We get solutions of x=2 AND x=2. X=2, when plugged back in is a valid solution. X must equal 2 which answers the questions "what is the value of x." Sufficient.
Statement (2) We get solutions of x=2 AND x=-12/7. Both solutions WORK this time. -12/7 is a VALID solution this time. Plug it back into the equation to verify. So we can't sufficiently answer the question "what is the value of x" because there are two possible values of x. Not sufficient.

A is the answer because Statement (1) sufficiently provides an answer to the question.
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by suzeemunkee » Sun Jul 11, 2010 5:55 pm
barcebal,

Ohhh.... thanks for the clarification! I see the mistake I made now. Really appreciate your help! :D

- Suzanne
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by Testluv » Mon Jul 12, 2010 12:07 am
Hi Suzee,

essentially, the first question is a "yes/no" question while the second is a "value" question. On the GMAT, you'll see about 15 DS, about 5 of which will be yes/no, and 10 of which will be value.

For yes/no questions, a range of values can qualify for sufficiency while in value questions, only one value is required for sufficiency.

Note that for the first question, you can arrive at the right answer more quickly using reasoning + algebra rather than just algebra.

Is x>0?

(1) |x + 3| = 4x - 3

Absolute value is always positive or zero. Thus, the left-hand side is positive or zero. Because the left must equal the right, the right hand side is positive or zero. Thus:

4x-3>= 0

x>=3/4

If x is always bigger than or equal to 3/4, then it is definitely positive, and so (1) is sufficient. The same reasoning applies to (2).
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by suzeemunkee » Mon Jul 12, 2010 9:11 pm
Testluv wrote:Hi Suzee,

essentially, the first question is a "yes/no" question while the second is a "value" question. On the GMAT, you'll see about 15 DS, about 5 of which will be yes/no, and 10 of which will be value.

For yes/no questions, a range of values can qualify for sufficiency while in value questions, only one value is required for sufficiency.

Note that for the first question, you can arrive at the right answer more quickly using reasoning + algebra rather than just algebra.

Is x>0?

(1) |x + 3| = 4x - 3

Absolute value is always positive or zero. Thus, the left-hand side is positive or zero. Because the left must equal the right, the right hand side is positive or zero. Thus:

4x-3>= 0

x>=3/4

If x is always bigger than or equal to 3/4, then it is definitely positive, and so (1) is sufficient. The same reasoning applies to (2).
Thanks, Testluv. I like the way you used logical reasoning to solve the yes/no DS. Appreciate your input!
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