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To determine which set of numbers could *not* represent the sides of a triangle, we need to apply the **triangle inequality theorem**. This theorem states that for any three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Let’s check each option:

1. **{7, 14, 18}**:  
   - \(7 + 14 = 21 > 18\)  
   - \(7 + 18 = 25 > 14\)  
   - \(14 + 18 = 32 > 7\)  
   This set satisfies the triangle inequality, so these can form a triangle.

2. **{7, 13, 22}**:  
   - \(7 + 13 = 20 \not> 22\)  
   Since 20 is not greater than 22, this set does not satisfy the triangle inequality, so these cannot form a triangle.

3. **{7, 18, 23}**:  
   - \(7 + 18 = 25 > 23\)  
   - \(7 + 23 = 30 > 18\)  
   - \(18 + 23 = 41 > 7\)  
   This set satisfies the triangle inequality, so these can form a triangle.

4. **{14, 18, 30}**:  
   - \(14 + 18 = 32 > 30\)  
   - \(14 + 30 = 44 > 18\)  
   - \(18 + 30 = 48 > 14\)  
   This set satisfies the triangle inequality, so these can form a triangle.

### Conclusion:
The set **{7, 13, 22}** does not satisfy the triangle inequality theorem and cannot represent the sides of a triangle.

Would you like more details on how the triangle inequality works or have any other questions?

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Here are 5 related questions to expand on this topic:

1. How does the triangle inequality theorem relate to the Pythagorean theorem?
2. Can a triangle have side lengths that are all equal? What type of triangle is this?
3. What are the properties of a right triangle in relation to side lengths?
4. How can you check if three sides form a valid triangle without calculating each inequality?
5. How does the triangle inequality theorem apply to angles in a triangle?

**Tip:** The triangle inequality theorem is a quick way to check the feasibility of triangle formation just by comparing the side lengths without needing to draw the figure.

Which of the following sets of numbers could not represent the three sides of a triangle?

Learn how to determine which sets of numbers can represent the sides of a triangle by applying the Triangle Inequality Theorem. This problem is suitable for students in Grades 6-9 and explains the necessary conditions for three sides to form a valid triangle.

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