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Answer :
Answer:
D. -3.0 < x < 7.5
Step-by-step explanation:
Given the side lengths of a triangle: 6.5 units, 3.5 units, and 2.5 + x units
For the triangle to exist:
- A side length cannot be negative
- A side length cannot be zero
Therefore, the third side length
x+2.5>0
x>-2.5
x must be greater than -2.5 and less than or equal to 6.5 units.
The correct option is D which satisfies this condition.
D. -3.0 < x < 7.5
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Final answer:
The given dimensions satisfy the Triangle Inequality Theorem to ensure the existence of a triangle if 0.5 < x < 7.5. So, the correct answer is A: 0.5 < x < 7.5.
Explanation:
A triangle can exist if and only if the sum of the lengths of any two sides is greater than the length of the third side. This is known as the Triangle Inequality Theorem. In this case, the three sides of the triangle are 6.5 units, 3.5 units, and 2.5 + x units. So, the sum of the lengths of two shortest sides (which are 2.5 + x units and 3.5 units) should be greater than the length of the largest side (which is 6.5 units). Similarly, the sum of the lengths of the other two sides should also be greater than the remaining side.
Therefore, our inequalities are 2.5 + x + 3.5 > 6.5 and 3.5 + 6.5 > 2.5 + x. Solving these inequalities yields the solution 0.5 < x < 7.5, so the correct option is A: 0.5 < x < 7.5.
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