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Is |x|+x <2?

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by Max@Math Revolution » Tue Jun 12, 2018 1:52 am

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Difficulty—

[GMAT math practice question]

$$Is\ |x|+x<2?$$

$$1)\ x>-1$$
$$2)\ x<0$$
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Source: — Data Sufficiency |
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by Max@Math Revolution » Thu Jun 14, 2018 1:33 am
=>

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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The definition of the absolute value gives us two cases to consider when examining the question.

Case 1: x ≥ 0
|x| + x < 2
=> x + x < 2
=> 2x < 2
=> x < 1
The question asks if 0 ≤ x < 1 in this case.

Case 2: x < 0
|x| + x < 2
=> (-x) + x < 2
=> 0 < 2
As this is always true, the answer is always "yes" if x < 0.

Combining these two cases shows that the question asks if x < 1.

In inequality questions, the law "Question is King" tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient

Thus condition 1) is not sufficient, but condition 2) is sufficient since the solution set of the question includes the solution set of condition 2), but it doesn't include that of condition 1).

Therefore, B is the answer.

Answer: B
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by Scott@TargetTestPrep » Thu Jun 14, 2018 9:51 am
Max@Math Revolution wrote:[GMAT math practice question]

$$Is\ |x|+x<2?$$

$$1)\ x>-1$$
$$2)\ x<0$$
Statement One Alone:

x > -1

If x = 0, then |x| + x = |0| + 0 = 0 < 2. However, if x = 2, then |2| + 2 = 4 is not less than 2. Statement one alone is not sufficient.

Statement Two Alone:

x < 0

If x = -1, then |x| + x = |-1| + (-1) = 0 < 2. If x = -2, then |-2| + (-2) = 0 < 2. It seems |-x| + x = 0 if x < 0. Let's prove this is indeed the case.

Since x < 0, |x| = -x, so |x| + x = -x + x = 0, which is always less than 2. Statement two alone is sufficient.

Answer: B

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