Statement 1 is not sufficient because x could also equal -4:

$$\sqrt{\left(-4\right)^2}=\sqrt{16}=4\ $$

Keep an eye out for negative options unless the question specifically states that x is positive! Similarly, look for non-integer options unless the question specifically states that the answer is an integer, etc.

As for why B is correct, knowing that the absolute value of x (which must be positive) is equal to negative x tells us that x itself must be negative. The negative of a negative number is positive

- (-n) = n

If x is a negative number, it cannot be a positive number, like 4. So this statement is sufficient to answer "no, x does not equal 4".

Hi M7MBA,

We're asked if X is equal to 4. This is a YES/NO question. We can solve it with a mix of 'math' and some Number Properties.

$$\left(1\right)\ \sqrt{x^2}=4.$$

According to the "Order of Operations" rules, we have to square the value of X BEFORE we square-root the result. Thus, there are 2 values that 'fit' the equation in Fact 1: +4 (which would give us a YES answer) and -4 (which would give us a NO answer).
Fact 1 is SUFFICIENT

$$\left(2\right)\ \left|x\right|=-x\ .$$

The result absolute value on the 'left' side of the equation can either be positive or 0. Since that value equals NEGATIVE X, then X must either be a NEGATIVE number or 0. By extension, it can NEVER be 4, so the answer to the question is ALWAYS NO.
Fact 2 is SUFFICIENT

Final Answer: B

GMAT assassins aren't born, they're made,
Rich