Answer:
The least integer x for which x, x^5, and 2x-15 may represent the lengths of the sides of a triangle is x = 4.
Explanation:
To determine the least integer x for which x, x^5, and 2x-15 may represent the lengths of the sides of a triangle, we need to apply the triangle inequality theorem.
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we have three sides: x, x^5, and 2x-15.
To satisfy the triangle inequality, we need to ensure that the sum of any two sides is greater than the length of the remaining side.
So, we have the following conditions:
x + x^5 > 2x-15,
x + (2x-15) > x^5,
x^5 + (2x-15) > x.
Simplifying these inequalities, we get:
x^5 - x - 15 > 0,
x^5 - 3x + 15 > 0,
x^5 - 3x - 15 > 0.
To find the least integer x that satisfies these conditions, we can use trial and error or solve the inequalities algebraically.
By testing different integer values, we find that the least integer x that satisfies all three inequalities is x = 4.
Therefore,
The least integer x that represent the lengths of the sides of a triangle is x = 4.