|x + 2| = |y + 2| What is the Value of x + y ? GMAT Data Sufficiency

Question: |x + 2| = |y + 2| what is the value of x + y ?

1. xy < 0
2. x > 2 and y < 2

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: D

Solution and Explanation:
Approach Solution 1:
The problem statement says |x + 2| = |y + 2|

We need to square both sides:

We get:

=> \(x^2\)+4x+4 = \(y^2\)+4y+4

=> \(x^2\) -\(y^2\)+4x-4y = 4-4

=> (x+y)(x-y) + 4(x-y) = 0

=> (x−y)(x+y+4)=0

Either x=y or x+y=−4
Now, we will consider the provided options and try to fit those in:

1. xy<0 --> the first case is not possible, since if x=y, then xy =\(x^2\), which is ≥0, not <0, as given in this statement, hence we have the second case: x+y=−4. This is Sufficient.
   Considering the second option x > 2 and y < 2
   This statement implies that x≠y, therefore x+y=−4. Sufficient.

Approach Solution 2:
For this problem, Variable approach is the easiest and quickest way to find the answer without actually solving the problem.
We must remember that an equal number of variables and independent equations ensures a solution.

Question: If |x+2|=|y+2|, what is the value of x+y?

Options:

1. xy<0
2. x>2 and y<2

In the original condition, the problem statement becomes x+2=±(y+2).
x=y or x+y=-4 are derived from x+2=y+2 or x+2=-(y+2).

Then, there are 2 variables(x,y) and 1 equation(|x+2|=|y+2|), which should match with the number of equations.
So we need 1 more equation. For 1) 1 equation, for 2) 1 equation,
For 1), since x≠y, x+y=-4, which is unique and sufficient.
For 2), since x≠y, x+y=-4, which is unique and sufficient.

Approach Solution 3:
Given: |x + 2| = |y + 2|
Key property: If |a| = |b|, then EITHER a = b OR a = -b
So, if |x + 2| = |y + 2|, then there are two possible cases:

case i: x + 2 = y + 2
Subtract 2 from both sides of the equation to get: x = y

case ii: x + 2 = -(y + 2)
Simplify: x + 2 = -y - 2
Subtract 2 from both sides of the equation to get: x = -y - 4
Add y to both sides to get: x + y = -4

Asked question: What is the value of x + y?

Statement 1: xy < 0
If the product xy is negative, we can conclude that one of the values (x or y) is POSITIVE, and the other value is NEGATIVE.
This means that x cannot equal y, which means case i cannot be true, which means case ii (x + y = -4) MUST BE TRUE
The answer to the target question is x + y = -4
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x > 2 and y < 2
Combine inequalities to get: y < 2 < x
This means that x cannot equal y, which means case i cannot be true, which means case ii (x + y = -4) MUST BE TRUE
The answer to the target question is x + y = -4
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

“|x + 2| = |y + 2| what is the value of x + y ?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of 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

Suggested GMAT Data Sufficiency Questions

• Each of the Students in a Certain Class Received a Single Grade of P GMAT Data Sufficiency
• Does The Graphical Representation Of The Quadratic Function GMAT Data Sufficiency
• What is the Value of m? (1) |m| = −36/m (2) 2m + 2|m| = 0 GMAT Data Sufficiency
• Do The Diagonals Of A Quadrilateral ABCD Bisect Each Other GMAT Data Sufficiency
• If X = 0.rstu, Where r, s, t, And u Each Represent A Nonzero Digit GMAT Data Sufficiency
• Is The Product Of a And b Equal To 1? GMAT Data Sufficiency
• Material A Costs $3 Per Kilogram, And Material B Costs $5 Per Kilogram GMAT Data Sufficiency
• Is the Product of a Positive and a Negative Integer Less Than -10? GMAT Data Sufficiency
• If k is not Equal to 0, 1, or -1 is 1/k>0? GMAT Data Sufficiency
• If x Is A Positive Integer, What Is The Value Of (x+24)^(½) − x^(½) GMAT Data Sufficiency
• Points x And y Lie On A Line Segment. How Many Units Long Is The Line GMAT Data Sufficiency
• Which Sport Utility Vehicle has a Higher List Price, the Touristo or GMAT Data Sufficiency
• What is The Average (arithmetic mean) Annual Salary of the 6 Employees GMAT Data Sufficiency
• If S is a Set of Ten Consecutive Integers, is the Integer 5 GMAT Data Sufficiency
• What is the Perimeter of a Square Chessboard, If a Chessboard is made GMAT Data Sufficiency
• Is the sum of four different positive prime integers an odd integer? GMAT Data Sufficiency
• Is Positive Integer x Divisible By 24? GMAT Data Sufficiency
• 10 Students took a Chemistry Exam that was Graded on a Dcale of 0 to 100 GMAT Data Sufficiency
• If m and n are positive integers, is the remainder of GMAT Data Sufficiency
• In quadrilateral ABCD, is AC longer than BD GMAT Data Sufficiency
• N is a positive integer. Is n the square of an integer? GMAT Data Sufficiency
• If z Is An Integer Smaller Than Integer n But Greater Than Zero GMAT Data Sufficiency
• If y Is An Odd Number And x^3 ∗ y^4 = 648, y=? 1) y Is Positive 2) x Is Gmat Data Sufficiency
• A Pyramid-Shaped Box To Protect A Plant Is Constructed With 4 Lateral Faces GMAT Data Sufficiency
• Is abc/d an Integer if a, b, c, and d are Positive integers? GMAT Data Sufficiency
• If n is a positive integer, is (1/10)^n < 0.01? GMAT Data Sufficiency
• If m and n are nonzero integers, is m^n an integer? GMAT Data Sufficiency
• In A Certain Office, 50 Percent Of The Employees Are College GMAT Data Sufficiency
• As Shown Above, A Rectangle Is Divided Into Smaller Rectangles Of The GMAT Data Sufficiency
• The Revenue of a Company is Directly Proportional to the Number of Customers GMAT Data Sufficiency
• Starting from the same point, a sparrow and a hawk flew in opposite directions GMAT Data Sufficiency
• Even/Odd: Are Negative Numbers and Decimals Even or Odd? GMAT Data Sufficiency
• If x,y and z are positive numbers, is z between x and y? GMAT Data Sufficiency
• What Is The Value Of 2^X + 2^(-X)? GMAT Data Sufficiency
• Is Quadrilateral ABCD a Square? GMAT Data Sufficiency
• Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 GMAT Data Sufficiency
• On a Trolley Ride Around an Amusement Park, a Child GMAT Data Sufficiency
• If x Is The Sum Of Factorials Of The First n Positive Integers GMAT Data Sufficiency
• Is Quadrilateral ABCD A Rhombus? GMAT Data Sufficiency
• What Is The Volume Of Rectangular Box R? GMAT Data Sufficiency
• On Sunday Morning, a Printing Press Printed its Newspapers at a GMAT Data Sufficiency
• Is n<0? GMAT Data Sufficiency
• Is n an even number GMAT Data Sufficiency
• If x ≠ 0, what is the value of x? GMAT Data Sufficiency
• If x is an integer, is 9^x + 9^(-x) = b ? (1) 3^x + 3^(-x) = (b + 2) GMAT Data Sufficiency
• If x is an integer, can the number (5/28)(3.02)(90%)(x) be GMAT Data Sufficiency
• Is quadrilateral ABCD a parallelogram? GMAT Data Sufficiency
• If Vertices of a Triangle have Coordinates (-2,2) (3,2) GMAT Data Sufficiency
• If Set S Consists of Even Number of Integer, Is the Median GMAT Data Sufficiency
• What Is The Value Of N GMAT Data Sufficiency