1) x ≤ y
This means that x could be equal to y, but it could also mean that x is less than y. The statement is therefore not sufficient.

2) |x| ≥ |y|
If x = y, the |x| would also be equal to |y|. Since |x| could be greater than |y|, this statement is not sufficient.

1) + 2) Together
This is not sufficient. Consider the scenario when x = -5 and y = 2, which satisfies both of the conditions of statements 1) and 2). However, so does the scenario when x = -5 and y = -5. Thus statements 1) and 2) can both be satisfies whether or not x = y.

The answer is (E).

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
If x =1, and y = 1, then x and y satisfy both conditions, and the answer is "yes" since x = y.
If x = -2, and y = 1, then x and y satisfy both conditions, but the answer is "no" since x ≠ y.

Thus, both conditions together are not sufficient, since they do not yield a unique solution.

Therefore, E is the answer.
Answer: E

Statement 1
$$x\le y$$
This means that x < y OR x = y, So I can conclude that
$$x=y\ or\ x\ne y$$
Information given is not enough to answer the question, hence statement is NOT SUFFICIENT.

Statement 2
$$\left|x\right|\ge\left|y\right|$$
This means that
$$\left|x\right|>\left|y\right|\ OR\ \left|x\right|=\left|y\right|$$
If x > y hence |x| > |y| but if x = y then |x| = |y|
Since there is no specific answer, information given is not enough to answer this question.
Hence statement 2 is INSUFFICIENT.

Combining both statements together
$$x\le y-----------statement\ 1$$
$$\left|x\right|\le\left|y\right|-----------statement\ 2$$
$$\left|x\right|\ge\left|y\right|\ where\ x\le y$$
Statement 1 and 2 both have more than one solution hence, both statements together are INSUFFICIENT.

$$answer\ is\ Option\ E$$