To determine the inequality that ensures the triangle exists, we can 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.

Given the side lengths:

• $AC = 18$ units
• $BC = 6x$ units
• $AB = 2x + 4$ units

We will apply the Triangle Inequality Theorem to each pair of sides:

1. $AB + BC > AC$ $(2x + 4) + 6x > 18$ $8x + 4 > 18$ $8x > 14$ $x > \frac{7}{4} \quad \text{or} \quad x > 1.75$

2. $AB + AC > BC$ $(2x + 4) + 18 > 6x$ $2x + 22 > 6x$ $22 > 4x$ $x < \frac{22}{4} \quad \text{or} \quad x < 5.5$

3. $BC + AC > AB$ $6x + 18 > 2x + 4$ $4x + 18 > 4$ 4x > -14 \quad \text{(which is always true for positive x)}

The valid values for $x$ must satisfy both $x > 1.75$ and $x < 5.5$. Therefore, the correct inequality that represents the values of $x$ ensuring the triangle exists is:

$1.75 < x < 5.5$

If this matches one of the answer choices (which seems to be missing), select that choice.

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Related Questions:

1. How do you use the Triangle Inequality Theorem with different types of triangles?
2. What happens if the side lengths violate the Triangle Inequality Theorem?
3. How would you solve for $x$ if additional conditions were placed on the sides?
4. Can the Triangle Inequality Theorem be applied in coordinate geometry?
5. How would this theorem apply if one or more side lengths were given as algebraic expressions?

Tip: Always verify that all inequalities derived from the Triangle Inequality Theorem are satisfied to ensure the triangle's existence.