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?

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.