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Is √(x+1)^2=x+1 ?

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Is √(x+1)^2=x+1 ?

by Max@Math Revolution » Mon Dec 31, 2018 1:11 am

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[Math Revolution GMAT math practice question]

Is $$\sqrt{\left(x+1\right)^2}=x+1$$ ?

1) x(x-2) = 0
2) x(x+2) = 0
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Source: — Data Sufficiency |
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by GMATGuruNY » Mon Dec 31, 2018 11:33 am

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Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

Is $$\sqrt{\left(x+1\right)^2}=x+1$$ ?

1) x(x-2) = 0
2) x(x+2) = 0
√(a²) = |a|.
Question stem, rephrased:
Is |x+1| = x+1?
The answer will be YES if x+1 is NONNEGATIVE:
x+1≥0
x≥-1.
Question stem, rephrased again:
Is x≥-1?

Statement 1:
x=0 or x=2.
Since x≥-1 in both cases, the answer to the question stem is YES.
SUFFICIENT.

Statement 2:
x=0 or x=-2.
If x=0, the answer to the question stem is YES, but if x=-2, the answer to the questions stem is NO.
INSUFFICIENT.

The correct answer is A.
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by Max@Math Revolution » Wed Jan 02, 2019 12:16 am

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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.

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

The question is equivalent to asking if x ≥ -1 as shown below:
$$\sqrt{\left(x+1\right)^2}=x+1$$
=> |x+1| = x+1
=> x ≥ -1

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
x(x-2) = 0
=> x = 0 or x = 2
If x = 0, then x ≥ -1 and the answer is 'yes'.
If x = 2, then x ≥ -1 and the answer is 'yes'.
Since it gives a unique answer, condition 1) is sufficient.

Condition 2)
x(x+2) = 0
=> x = 0 or x = -2
If x = 0, then x ≥ -1 and the answer is 'yes'.
If x = -2, then x < -1 and the answer is 'no'.
Since it does not give a unique answer, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

If the original condition includes "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations" etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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