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Inequalities with Absolute value

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Inequalities with Absolute value

by aditya.j » Wed Apr 11, 2012 6:46 pm
This is a question from Knewton Prep:

[On the diagram a is a point between 0 and 2, approx. 1.75]

On the number line above, a is the coordinate of the point shown. Which of the following must be true?

|a| < 2
|a - 3| > 1
|a| + |a - 2| = 2

(A)

I only
(B)

I and II only
(C)

I and III only
(D)

II and III only
(E)

[spoiler] I was particularly confused with pt. 2---> |a - 3| > 1

I tried solving it and found out that a >4, a<2. At this point, i thought that it wasn't true, cos a is definitely less than 2.. I think i am missing something here...Need some help here..

[/spoiler]

The OA is E
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by Jim@StratusPrep » Wed Apr 11, 2012 6:50 pm
If a is less than 2 then a - 3 < 1 or |a-3| > 1
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by spider » Thu Apr 12, 2012 4:22 am
Jim@StratusPrep wrote:If a is less than 2 then a - 3 < 1 or |a-3| > 1
thank you sir!

But why I is not true too? Thanks in advance.
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by Stuart@KaplanGMAT » Thu Apr 12, 2012 9:48 am
aditya.j wrote:This is a question from Knewton Prep:

[On the diagram a is a point between 0 and 2, approx. 1.75]

On the number line above, a is the coordinate of the point shown. Which of the following must be true?

|a| < 2
|a - 3| > 1
|a| + |a - 2| = 2

(A)

I only
(B)

I and II only
(C)

I and III only
(D)

II and III only
(E)
Hello!

If a does in fact equal 1.75, then of course |1.75| < 2 and (I) must be true. Are you sure that you've accurately reproduced the question/answers?

We can solve very quickly just by plugging in a=1.75 to each statement:

(I) |1.75|<2... Is 1.75 < 2? YES, therefore TRUE
(II) |1.75 - 3| = |-1.25| = 1.25... Is 1.25 > 1? YES, therefore TRUE.
(III) |1.75| + |1.75 - 2| = 1.75 + |-.25| = 1.75 + .25 = 2... is 2 = 2? YES, therefore TRUE.

Accordingly, the correct answer is I, II and III, which I imagine is what (E) actually says (you didn't post all of option (E)).
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