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Text Explanation

This problem refers to a geometric theorem known as the Triangle Inequality. One way to say the Triangle Inequality is: the sum of any two sides of a triangle is greater than the third. Another perfectly valid way to say this is: any side of a triangle is greater than the difference of the other two sides and less than the sum of the other two sides. This latter statement will help us here. For more on the Triangle Inequality, see this GMAT blog post.

Let's call the third side L.

From the second statement of the triangle inequality, we know

(3x + 4) – (2x + 1) < L < (3x + 4) + (2x + 1)

x + 3 < L < 5x + 5

Clearly (x + 2) is always less than (x + 3), so choices B is wrong.

Clearly (5x + 6) is always more than (5x + 5), so choice D is wrong.

For the other choices, we have to be careful. Everything here is in variable form, and we are asked to find all POSSIBLE lengths, so if any value of x, any value at all, makes the expression work in the inequality above, then the expression works as a "possible" length. The algebraic expression does not need to work at all values of x; rather, it only needs to work at some or even just one value of x in order for that expression to be a "possible" length. The only restriction we have is

Would (4x + 5) be possible? Let's try different values.

x = 1 gives us 4 < 9 < 10, which is possible, so A is a possible length.

Would (6x + 1) be possible? Let's try different values.

x = 1 gives us 4 < 7 < 10, which is possible, so C is a possible length.

Would (2x + 17) be possible? Let's try different values.

x = 1 gives us 4 < 19 > 10 ---- no good.

x = 2 gives us 5 < 21 > 15 ---- no good.

x = 3 gives us 6 < 23 > 20 ---- no good.

x = 4 gives us 7 < 25 = 25 ---- no good. (Equal does not work)

x = 5 gives us 8 < 27 < 30, which is possible, so E is a possible length.

All we needed was one possible value of x that makes the inequality true.

Choices A & C & E are correct.

FAQ: How can choice (C) be the answer? If x = 4 or 5, it does not hold true.

A: The key part of the question is "which of the following could be the length of the third side of the triangle." For this type of question, we just need one scenario in which choice (C) works, and then it could be true. If the question asked which must be true, then you would be right that (C) would not work. Since x = 1 works, this is a possible side of the triangle.

FAQ: Answer choice (E) 2x + 17 is larger than 5x + 5 if you assume x = 2. So that cannot be solution? Can it?

A: I can see why you might think 2x + 17 cannot be an answer. It certainly doesn't work when you plug in smaller numbers for x. However, it does hold true when you plug in larger numbers. Take, for example, x = 10. When you plug them into their respective equations, you get 13 < 37 < 55, which works. Remember that we only need one case to work in order for the answer choice to be possible