You could solve this algebraically, but it might be faster and easier to just use some picked numbers along with logic.
For each of the statements if there is a way to make |a-3| less than and also greater than or equal to 2, then the statement is insufficient.
Statement 1 gives us a set of numbers to work with, all those greater than 2 and less than -2.
If we plug 3 into the original statement, we get |3-3| = |0| < 2.
If we plug -3 into the original statement, we get |-3-3| = |-6| > 2.
So Statement 1 is insufficient.
Statement 2 gives us another set of numbers, all those within 3 units of 1.
3 works again, and once again gives us |0| < 2.
-1 works, and plugged into the original statement -1 gives us |-1-3| = |-4| > 2.
So Statement 2 is insufficient.
Let's try combining the statements.
Given Statement 1, a cannot be 0 and if a is negative, it has to be less than -2.
Given Statement 2, if a is negative, it has to be greater than -2.
So no negative values of a are possible.
Given Statement 1, if a is positive, it has to be greater than 2.
Given Statement 2, a has to be be within 3 units of 1, meaning a < 4.
So with the statements combined the only possible values of a are 2 < a < 4.
Let's test the end points.
Plugging 4 into the original equation gives us |4-3| = |1| < 2
Plugging 2 into the original equation gives us |2-3| = |-1| < 2
Any a between 4 and 2 will give us a-3 between 1 and -1.
So combined the statements are sufficient and the answer is C.
AS # 32
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Source: Beat The GMAT — Data Sufficiency |
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|x-y| = the DISTANCE between x and y.oquiella wrote:32. Is |a-3| < 2?
(1) |a| > 2
(2) |a-1| < 3
|x| = the distance between x and 0.
|a-3| < 2?
In words:
Is the distance between a and 3 less than 2?
Put another way:
On the number line, is a within the red portion below?
.....1<-----3----->5.....
Question stem, rephrased:
On the number line, is a between 1 and 5?
Statement 1: |a| > 2
In words:
The distance between a and 0 is is greater than 2.
Put another way:
On the number line, a is within one of the two blue portions below:
<----- -2.....0.....2 ----->
If a=3, then a is between 1 and 5 on the number line.
If a=-3, then a is NOT between 1 and 5 on the number line.
INSUFFICIENT.
Statement 2: |a-1| < 3
In words:
The distance between a and 1 less than 3.
Put another way:
On the number line, a is within the green portion below:
.....-2<-----1----->4.....
If a=3, then a is between 1 and 5 on the number line.
If a=0, then a is NOT between 1 and 5 on the number line.
INSUFFICIENT.
Statements combined:
<----- -2.....0....2 ----->
.............-2<---1--->4.....
The colored ranges overlap between 2 and 4.
Implication:
To satisfy both statements, a must be within the red range below:
......2<----->4......
Thus, a must be between 1 and 5 on the number line.
SUFFICIENT.
The correct answer is C.
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My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
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