One important thing to know about the square root of a square term:

(1) If it's a number under the square root sign, the answer is only the positive root: sqrt[(-3)^2] = sqrt[(3)^2] = sqrt[9] = 3.

(2) If there are variables under the square root sign, you must consider the fact that the squared "thing" may have been either pos or neg to begin with: sqrt[(-x)^2] = sqrt[(x)^2] = |x|. We are still looking for the positive root, but we don't know whether +x or -x is actually greater than zero--that depends on the sign of x.

So for this question: Is sqrt[(x-3)^2] = 3-x?

Rephrase to: Is |x-3| = 3-x?

If stuff in the absolute value sign is positive or zero, this becomes:

Is x-3 = 3-x?

Is 2x = 6?

Is x = 3?

If stuff in the absolute value sign is negative, this becomes:

Is -(x-3) = 3-x?

Is -x+3 = 3-x?

The answer is yes for all x, but remember this was just "all x" such that x-3 was negative, or x < 3: So we ask "Is x < 3?"

Put the two cases together for the final rephrase:

"Is x =<3 ? "(1) x is not 3, but no info on whether it is less than or greater than 3. INSUFF.

(2) -x|x| > 0.

|x| is positive, so -x would have to be positive too (pos*pos > 0, but neg*pos < 0). Thus, x is negative. If x is negative, it is definitely less than 3. The answer is definitely Yes. SUFF.

The answer is B.

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Emily Sledge | Manhattan GMAT Instructor | St. Louis

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