What is the Value of y? GMAT Data Sufficiency

Question: What is the value of y?

(1) \(3|x^2 - 4| = y - 2\)

(2) \(|3 - y| = 1\)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: C

Solution and Explanation:
Approach Solution 1:
The problem statement asks to find the value of y.

Statement 1:  \( 3 |x^2−4| = y−2\)
In this equation, since the left-hand side of this expression is  \( 3 |x^2−4| \) which is an absolute value, it can't be negative. Therefore, the right-hand side of the expression (y −2 ) must also be non-negative. That is, it implies \(y − 2 ≥ 0\) which means \(y ≥ 2\)

Solving the equation \(3 |x2−4| = y−2\) further:
Since it is asked in the question to find the value of y, from this statement we can infer only that y ≥ 2. This is because LHS is an absolute value and it is never negative. Hence RHS also cannot be negative. Therefore, Not sufficient.

Statement 2: \(| 3 − y | = 11\)

\(y < 3 \)

=> 3 − y = 11

=> y = - 8

\(y ≥ 3 \)

=> 3 + y = 11

=> y = 14

Therefore, we are getting two values for y.
Hence, Not sufficient.

Combining both statements (1)+ (2) we get:
From statement 1 we get: y ≥ 2,
From statement 2, we get: y = -8,14.

Therefore, from both statements, we can say, the value of y is 14.
Hence, both statements TOGETHER are sufficient.

Approach Solution 2:
The problem statement asks to find the value of y.

Statement 1: 3|x² – 4| = y – 2

It is required to choose any value of x, and we will find a corresponding value of y.

Let’s consider these two possible cases:
Case A:
If x = 1, which means y = 11 (plugging x = 1 into the equation, we get the value of y).

In this case, the answer to the question is y = 11
Case B:
If x = 2, which means y = 2.

In this case, the answer to the question is y = 2
Since we cannot answer the question with certainty, statement 1 is NOT SUFFICIENT.

Statement 2: |3 – y| = 11

Useful property: If |x| = k, then x = k or x = -k
So, either 3 – y = 11 or 3 – y = -11

Let's examine each case:
Case A: If 3 – y = 11, then y = -8.

In this case, the answer to the question is y = -8
Case B: If 3 – y = -11, then y = 14.

In this case, the answer to the target question is y = 14
Since we cannot answer the question with certainty, statement 2 is NOT SUFFICIENT.

Combining Statements 1 and 2, we get: 
Statement 2 tells us that EITHER y = -8 OR y = 14
There are only two possible values of y.
At this point, we need only determine whether each of these values of y will yield an actual solution for the statement 1 equation (3|x² – 4| = y – 2 )

Case A: y = -8, which means we get: 3|x² – 4| = (-8) – 2
Simplify: 3|x² – 4| = -10
Therefore, we get: |x² – 4| = -10/3
Since the absolute value of an expression is always greater than or equal to 0, we can conclude that this equation has ZERO solutions.

At this point, we can see that y = -8 is not a proper solution. This means it MUST be the case that y = 14. This implies that the COMBINED statements are sufficient.

However, let's see by examining y = 14.
Case B: y = 14, which means we get: 3|x² – 4| = 14 – 2
Simplify: 3|x² – 4| = 12
Divide both sides by 3 to get: |x² – 4| = 4
From the above property, we can conclude that EITHER x² – 4 = 4 OR x² – 4 = -4

If x² – 4 = 4, then x² = 8, so x = √8 OR x = -√8
In other words, one possible solution is x = √8 and y = 12
Another possible solution is x = -√8 and y = 12

Similarly, if x² – 4 = -4, then x² = 0, which means x = 0
So another possible solution is x = 0 and y = 12

Notice that, for ALL THREE possible solutions, the answer to the question is always the same: y = 12
So, it must be the case that y = 12
Since we can answer the question with certainty, the combined statements are SUFFICIENT.

Approach Solution 3:
The problem statement asks to find the value of y.

Statment 1: \( 3 |x^2−4| = y−2\)
y = \( 3 |x^2−4| + 2\)

Case 1:
\(x^2−4 ≥ 0\) 
=> \(x^2 ≥ 2\)
=> \(x ≤ -2\) or \(x ≥ 2\)

y = \(3 (x^2 + 4) + 2\) = \(- 3 x^2 + 14\)

Case 2:
\(x^2−4 < 0\)
=>\(x^2 < 4\)
=> -2 < x < 2
y =\(3 (-x^2 + 4) + 2\) = \(-3x^2 + 14\)

Since the value of y is dependent on the value of x, therefore statement 1 is Insufficient.

Statement 2: |3 – y| = 11
=> |y - 3| = 11

Case 1:
y - 3  ≥ 0
=> y  ≥  3
y - 3 = 11
=> y = 14

Case 2:

y - 3 < 0
=> y < 3
-y + 3 = 11
=> y = -8

Therefore, we get 2 values of y. Hence statement 2 is Insufficient.

Adding Statement 1 and Statement 2, we get:
y can only be -8 or 14.
If x = 0, (i.e. the value in case 2 from statement 1):
y = \(-3x^2 + 14\) = 14

Therefore, the combined statements are SUFFICIENT.

“What is the value of y?”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Official Advanced Questions”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions include a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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