Full solution

Q. Which of the following sets of numbers could represent the three sides of a triangle?

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\{4,10,14\}

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\{12,25,36\}

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\{6,8,14\}

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\{15,30,46\}

Recall Triangle Inequality Theorem: Recall the Triangle Inequality Theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Apply Theorem to Set $\{4,10,14\}$ : Apply the Triangle Inequality Theorem to the first set $\{4,10,14\}$ . Check if 4 + 10 > 14, 4 + 14 > 10, and 10 + 14 > 4. $4 + 10 = 14$ which is not greater than $14$ , so the first set does not satisfy the Triangle Inequality Theorem.
Apply Theorem to Set $\{12,25,36\}$ : Apply the Triangle Inequality Theorem to the second set $\{12,25,36\}$ . Check if 12 + 25 > 36, 12 + 36 > 25, and 25 + 36 > 12. $12 + 25 = 37$ which is greater than $36$ , $12 + 36 = 48$ which is greater than $25$ , but $25 + 36 = 61$ which is not greater than $\{12,25,36\}$ $0$ , so the second set does not satisfy the Triangle Inequality Theorem.
Apply Theorem to Set $\{6,8,14\}$ : Apply the Triangle Inequality Theorem to the third set $\{6,8,14\}$ . Check if 6 + 8 > 14, 6 + 14 > 8, and 8 + 14 > 6. $6 + 8 = 14$ which is not greater than $14$ , so the third set does not satisfy the Triangle Inequality Theorem.
Apply Theorem to Set $\{15,30,46\}$ : Apply the Triangle Inequality Theorem to the fourth set $\{15,30,46\}$ . Check if 15 + 30 > 46, 15 + 46 > 30, and 30 + 46 > 15. $15 + 30 = 45$ which is not greater than $46$ , so the fourth set does not satisfy the Triangle Inequality Theorem.