To ensure that triangle ABC exists, we need to consider the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's assume that the lengths of the sides of triangle ABC are represented by the variables a, b, and c.
According to the options provided, the inequality that represents the values of x that ensure triangle ABC exists is:
C. 1 < x < 5
This is because if we substitute x with values within this range, the resulting lengths of the sides will satisfy the triangle inequality theorem.
To prove that the inequality 1 < x < 5 ensures the existence of triangle ABC, we need to show that for any value of x within this range, the lengths of the sides of triangle ABC satisfy the triangle inequality theorem.
The triangle inequality theorem states that for any triangle with side lengths a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side. In other words, for triangle ABC, we have:
a + b > c
b + c > a
c + a > b
Let's consider the inequality 1 < x < 5. This means that x is greater than 1 and less than 5.
To prove that triangle ABC exists for this range of x, we need to show that the lengths of the sides a, b, and c satisfy the triangle inequality theorem.
Let's consider the lengths of the sides in terms of x:
Side a = 2x + 4
Side b = x + 18
Side c = 6x
We will check if the inequalities hold for these side lengths:
a + b > c:
(2x + 4) + (x + 18) > 6x
3x + 22 > 6x
22 > 3x
b + c > a:
(x + 18) + 6x > 2x + 4
7x + 18 > 2x + 4
5x > -14
c + a > b:
6x + (2x + 4) > x + 18
8x + 4 > x + 18
7x > 14
From these inequalities, we can see that for any value of x within the range 1 < x < 5, the side lengths satisfy the triangle inequality theorem. Therefore, triangle ABC exists when 1 < x < 5.
This completes the proof that the inequality 1 < x < 5 ensures the existence of triangle ABC.