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If x is a positive integer, what is the value of \(\sqrt{x+24}-\sqrt{x}\)?

(1) \(\sqrt{x}\) is an integer (2) \(\sqrt{x+24}\) is an integer

We are given that x is a positive integer and must determine the value of √(x+24) - √x.

Statement One Alone:

√x is an integer.

Using the information in statement one, we could obtain multiple values for √(x+24) - √x. If x = 1, then √(x+24) - √x = √25 - √1 = 5 - 1 = 4; however, if x = 4, then √(x+24) - √x = √28 - √4 = √28 - 2. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

√(x+24) is an integer.

Using the information in statement two, we could obtain multiple values for √(x+24) - √x. If x = 1, then √(x+24) - √x = √25 - √1 = 5 - 1 = 4; however, if x = 12, then √(x+24) - √x = √36 - √12 = 6 - √12. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using the information in statements one and two, we still could obtain multiple values for √(x+24) - √x. If x = 1, then √(x+24) - √x = √25 - √1 = 5 - 1 = 4; however, if x = 25, then √(x+24) - √x = √49 - √25 = 7 - 5 = 2.

Answer: E
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If x is a positive integer, what is the value of \(\sqrt{x+24}-\sqrt{x}\)?

(1) \(\sqrt{x}\) is an integer (2) \(\sqrt{x+24}\) is an integer

Target question:What is the value of √(x + 24) - √x?

Given: x is a positive integer

Statement 1: √x is an integer There are several values of x that satisfy statement 1. Here are two: Case a: x = 1 (notice that √1 is an integer). In this case, √(x + 24) - √x = √(1 + 24) - √1 = 5 - 1 = 4 Case b: x = 4 (notice that √4 is an integer). In this case, √(x + 24) - √x = √(4 + 24) - √4 = √28 - 2 Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: √(x + 24) is an integer There are several values of x that satisfy statement 2. Here are two: Case a: x = 1 (notice that √(1 + 24) = √25 = 5, which is an integer). In this case, √(x + 24) - √x = √(1 + 24) - √1 = 5 - 1 = 4 Case b: x = 25 (notice that √(25 + 24) = √49 = 7, which is an integer). In this case, √(x + 24) - √x = √(25 + 24) - √25 = 7 - 5 = 2 Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined Case a: x = 1. In this case, √(x + 24) - √x = √(1 + 24) - √1 = 5 - 1 = 4 Case b: x = 25. In this case, √(x + 24) - √x = √(25 + 24) - √25 = 7 - 5 = 2 Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

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