Question

Find the range of possible measures of x if each set of expressions represents measures on the sides of a triangle. x+1, 5, 7

Answer 1

The range for x, to satisfy the conditions of triangle side lengths x+1, 5, and 7, is more than 1 and less than 11.

Step-by-step explanation:

To find the range of possible measures of x for the sides of a triangle with lengths x+1, 5, and 7, we use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

First, we find the minimum value of x by setting the sum of the two shortest sides equal to the longest side:

1. x + 1 + 5 = 7
2. x + 6 = 7
3. x = 7 - 6
4. x = 1 (This is not possible, as the sum must be greater).

Therefore, we add a small value, such as 0.1, to ensure the sum is greater, which would give us x > 1. At minimum, x must be greater than 1.

Next, we find the maximum value of x by setting the sum of the longest side and the unknown side greater than the other side:

1. 7 + x + 1 > 5
2. x + 8 > 5
3. x > 5 - 8
4. x > -3 (This is unnecessary since a side length cannot be negative).

However, for the sake of a triangle, x cannot be so large that x+1 would be greater than the sum of the other two sides, that is:

1. x + 1 < 5 + 7
2. x + 1 < 12
3. x < 12 - 1
4. x < 11

Thus, combining these inequalities, we find that the range for x is 1 < x < 11.