Okay, let's see. The problem is asking for an integer x that satisfies the inequality 1/4 < x/7 < 1/3. Hmm, so I need to find an integer x such that when you divide it by 7, the result is between 1/4 and 1/3. Alright, let me think about how to approach this.

First, maybe I should convert the fractions to decimals to get a better sense of the range we're dealing with. 1/4 is 0.25, and 1/3 is approximately 0.3333. So the inequality is 0.25 < x/7 < 0.3333. To find x, I can multiply all parts of the inequality by 7 to eliminate the denominator. Let me do that.

Multiplying the left part by 7: 0.25 * 7 = 1.75. Then the middle part is x. The right part: 0.3333 * 7 ≈ 2.3331. So the inequality becomes 1.75 < x < approximately 2.3331. Since x has to be an integer, let's list the integers between 1.75 and 2.3331. The integers in that range are 2. So x must be 2. Let me check that.

If x is 2, then 2/7 is approximately 0.2857. Is that between 0.25 and 0.3333? Yes, because 0.25 < 0.2857 < 0.3333. So that works. But wait, is there another integer between 1.75 and 2.3331? Let's see. The next integer after 1.75 is 2, and the next integer after 2.3331 is 3. So 2 is the only integer in that interval. Therefore, x must be 2. Let me verify once more.

Alternatively, maybe I should solve the inequalities step by step. Starting with 1/4 < x/7. Multiplying both sides by 7 gives 7*(1/4) < x. 7/4 is 1.75, so x > 1.75. Then, from x/7 < 1/3, multiplying both sides by 7 gives x < 7/3. 7 divided by 3 is approximately 2.3333. So x must be greater than 1.75 and less than 2.3333. Since x is an integer, the only possible value is 2. That confirms it again. So x is 2. I think that's the correct answer. Let me just make sure there's no other integer that could fit. If x were 1, then 1/7 is about 0.1429, which is less than 0.25. If x were 3, then 3/7 is approximately 0.4286, which is more than 0.3333. So yes, 2 is the only integer that works. Therefore, the answer is 2.
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