Answer :
The inequality that represents the values of x that ensure triangle ABC exists is A. 0.5 < x < 7.5
What is inequality?
Inequality is defined as the relation which makes a non-equal comparison between two given functions.
Given that the side lengths of a triangle are as; 6.5 units, 3.5 units, and 2.5 + x units
By using the property, the Sum of two sides is always greater than third side of triangle, then;
6.5 + 3.5 > 2.5 +x
=>10 >2.5 + x
=>2.5 + x< 10
=> x< 10 -2.5
=> x < 7.5
Also, by using the property, Difference of two sides is always lesser than third side of triangle then we get;
6.5 - 3.5 > 2.5 +x
=>3.0 > 2.5 +x
=> 2.5 +x > 3.0
=> x > 0.5
Now combining both we have;
. 0.5 < x < 7.5
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Final answer:
The correct inequality representing the values of x that ensure triangle ABC exists is 0.5 < x < 7.5. We derive this from applying the triangle inequality theorem to the given sides of the triangle.
Explanation:
To determine the range of values for x that ensure triangle ABC exists when given side lengths of 6.5 units, 3.5 units, and 2.5 + x units, we can use the triangle inequality theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Applying this theorem, we have two inequalities:
- 6.5 + 3.5 > 2.5 + x, which simplifies to 10 > 2.5 + x, or x < 7.5 after subtracting 2.5 from both sides.
- 6.5 + (2.5 + x) > 3.5, which simplifies to 9 + x > 3.5, or x > -5.5.
However, the third inequality comes from the fact that 3.5 + (2.5 + x) must be greater than 6.5, leading to x > 0.5.
Combining the inequalities, we get 0.5 < x < 7.5, which matches choice A.
