Postal wrote:

Is \(|2a – b| < 7\) ?

(1) \(2a – b < 7\)

(2) \(a = b + 3\)

OFFICIAL SOLUTION

Absolute value can be understood as distance from zero on the number line. The quantity 2 a – b has an absolute value less than 7 if (and only if) that quantity is less than 7 units away from 0 on a number line.

(1) INSUFFICIENT: This statement tells us that 2 a – b is less than 7, but this does not tell us whether it is less than 7 units away from 0. For instance, 2 a – b could be equal to –20.

(2) INSUFFICIENT: We can manipulate the equation a = b + 3 so that it tells us the value of 2 a – b:

a = b + 3

2 a = 2 b + 6

2 a – b = b + 6

Since there is no restriction on the value of b, b + 6 can be anywhere on the number line. This implies that 2 a – b can be anywhere on the number line, making both “yes” and “no” answers possible.

(1) & (2) INSUFFICIENT: Statement (2) does not restrict the value of 2 a – b at all, so combining it with statement (1) yields the same result as for statement (1) alone. In a sense, the catch in this problem is that there is no catch: we should be suitably suspicious when a problem seems easier than it ought to be, but we should then trust our analysis and choose confidently.

The correct answer is E.

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