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Any help, please?

by fambrini » Tue Nov 08, 2016 4:43 pm
If |2x + 1| <= 0, then

A) x must be less than or equal to - 1/2
B) x must be equal to - 1/2
C) x must be larger than - 1/2
D) x could be 1/2
E) x doesn't exist

OA: B

I can develop the inequality in order to reach x <= - 1/2 or x >= - 1/2. I just don't understand the official answer.

Thanks,
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by GMATGuruNY » Tue Nov 08, 2016 4:48 pm
fambrini wrote:If |2x + 1| <= 0, then

A) x must be less than or equal to - 1/2
B) x must be equal to - 1/2
C) x must be larger than - 1/2
D) x could be 1/2
E) x doesn't exist

OA: B

I can develop the inequality in order to reach x <= - 1/2 or x >= - 1/2. I just don't understand the official answer.

Thanks,
Fambrini
An absolute value can never be negative.
Thus, it is not possible that |2x+1| < 0.
It is only possible that |2x+1| = 0.
|2x+1| = 0 if 2x+1 = 0:
2x+1 = 0
2x = -1
x = -1/2.

The correct answer is B.
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by [email protected] » Wed Nov 09, 2016 10:36 am
Hi fambrini,

Before working through any of the 'steps' in this question, you should note the main 'concept' involved:

We're dealing with an Absolute Value - and the result of any Absolute Value calculation can NEVER be negative. It can be 0 or it can be positive. Thus, when we're given the following inequality...

|2X + 1| <= 0

... we should note the fact that the absolute value is 'less than or equal to 0.' Mathematically, it CANNOT be less than 0, so it MUST EQUAL 0. Knowing that, we have just one small calculation to get to the correct answer...

2X + 1 = 0
2X = -1
X = -1/2

Final Answer: B

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by Jeff@TargetTestPrep » Fri Nov 11, 2016 7:09 am
fambrini wrote:If |2x + 1| <= 0, then

A) x must be less than or equal to - 1/2
B) x must be equal to - 1/2
C) x must be larger than - 1/2
D) x could be 1/2
E) x doesn't exist

OA: B
This is an interesting question because the absolute value of anything cannot ever be LESS THAN ZERO. Thus, we can rewrite the equation as |2x + 1| = 0 and solve for x. We note that the absolute value of an expression can only equal zero if the expression itself is equal to zero.

2x + 1 = 0

2x = -1

x = -(1/2)

Answer: B

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by Brent@GMATPrepNow » Fri Nov 11, 2016 7:25 am
fambrini wrote:If |2x + 1| < 0, then

A) x must be less than or equal to - 1/2
B) x must be equal to - 1/2
C) x must be larger than - 1/2
D) x could be 1/2
E) x doesn't exist
When solving inequalities involving ABSOLUTE VALUE, there are 2 things you need to know:
Rule #1: If |something| < k, then -k < something < k
Rule #2: If |something| > k, then EITHER something > k OR something < -k
Note: these rules assume that k is not negative

So, when we apply rule #1, -0 < 2x + 1 < 0
That's odd!
2x + 1 is greater than or equal to 0 AND less than or equal to 0
So, it MUST be the case that 2x + 1 = 0
Solve to get x = 1/2
Answer: B

RELATED VIDEO
- Inequalities and absolute value: https://www.gmatprepnow.com/module/gmat ... /video/985
Brent Hanneson - Creator of GMATPrepNow.com
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by Matt@VeritasPrep » Fri Nov 11, 2016 2:09 pm
Let's do it conceptually.

|a - b| = 0 means "the distance from a to b is 0".

|a - b| ≤ 0 means "the distance from a to b is LESS THAN or EQUAL TO 0".

Obviously you can't have a distance that's less than 0, so

|2x + 1| ≤ 0 must really be

|2x + 1| = 0

and from there the problem is a cinch!

|2x + 1| = 0 means "the distance from 2x to -1 is 0", or 2x = -1, or x = -1/2.
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by Matt@VeritasPrep » Fri Nov 11, 2016 2:10 pm
And not to editorialize too much, but the CONCEPT of absolute value as distance is something I think more students ought to learn and remember: it can rescue you from some pretty thick algebraic fog!
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