If set S Consists of all Different Solutions of Equation |x – 4| = x, What is the Range of Set S GMAT Problem Solving

Question: If set S consists of all different solutions of equation |x – 4| = x, what is the range of set S?

1. 0
2. 2
3. 4
4. 6
5. 8

Correct Answer: A

Solution and Explanation:
Approach Solution 1:
If x is greater than or equal to 4, equation |x-4| = x turns into x – 4 = x. The last one has no solutions. If x is less than 4, equation |x-4| = x turns into 4-x = x or x = 2

Thus, set S has only one element: 2. The range of any one element set is 0.

Approach Solution 2:
If x > 4
x – 4 = x
Hence no solution
0 < x < 4
4 – x = x
x = 2
If x < 0
4 – x = -x

No solution
So x = 2 is only solution
So S = [2]
Range = 0

Approach Solution 3:
If there are 2 elements
Let 2 numbers be -1 and 5
Then range is 5-(-1) =6
Let 2 numbers -1 and -2
Range will be -1-(-2) = 2-1=1
If there are 2,2 then range is 0
Range is always >=0

“If set S consists of all different solutions of equation |x – 4| = x, what is the range of set S?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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