A triangle has sides of lengths 3x, 2x - 2, and 5x. If the perimeter of the triangle is 28, what is the length of the longest side?
(A) 3

(B) 4

(C) 9

(D) 10

(E) 15

To determine the length of the longest side of the triangle, we first need to set up and solve the given equation for the perimeter.

The sides of the triangle are:
- \( 3x \)
- \( 2x - 2 \)
- \( 5x \)

The perimeter of the triangle is given as 28, so we can write the equation:
\[
3x + (2x - 2) + 5x = 28
\]

Simplify the equation:
\[
3x + 2x - 2 + 5x = 28
\]

Combine like terms:
\[
10x - 2 = 28
\]

Add 2 to both sides to isolate the term with \( x \):
\[
10x = 30
\]

Divide both sides by 10 to solve for \( x \):
\[
x = 3
\]

Now, substitute \( x = 3 \) back into the expressions for the sides to find their lengths:
- \( 3x = 3 \cdot 3 = 9 \)
- \( 2x - 2 = 2 \cdot 3 - 2 = 6 - 2 = 4 \)
- \( 5x = 5 \cdot 3 = 15 \)

The lengths of the sides are 9, 4, and 15. Therefore, the longest side of the triangle is 15.

Thus, the correct answer is:
\[
\boxed{15 \text{ (E)}}
\]

Would you like more details or have any questions? Here are 5 additional questions you might find interesting:

1. How do you determine if three given lengths can form a triangle?
2. What is the Pythagorean theorem and how is it used?
3. How do you find the area of a triangle given its side lengths?
4. What are the properties of special triangles, such as isosceles and equilateral triangles?
5. How do you solve for unknown angles in a triangle?

**Tip:** When solving for variables in geometry problems, always check if the solution satisfies all given conditions, such as side lengths and angles fitting within the properties of the shapes.

Learn how to find the length of the longest side of a triangle with sides 3x, 2x - 2, and 5x, given that the perimeter is 28. This problem involves setting up and solving equations, applying triangle properties, and determining side lengths. Suitable for high school students.

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