To find the length of \( x \) in the big triangle, we can use the concept of similar triangles. The sides of similar triangles are proportional. Therefore, we can set up a proportion based on the corresponding sides of the small triangle and the big triangle.
In the small triangle:
- One side measures \( 11.4 \) inches.
- Another side measures \( 18 \) inches.
In the big triangle:
- The side corresponding to \( 11.4 \) inches is \( x \).
- The side corresponding to \( 18 \) inches is \( 23.4 \) inches.
We can set up the following proportion:
\[ \frac{x}{11.4} = \frac{23.4}{18} \]
Now we will solve for \( x \). First, cross-multiply:
\[ x \cdot 18 = 11.4 \cdot 23.4 \]
Calculating \( 11.4 \cdot 23.4 \):
\[ 11.4 \cdot 23.4 = 267.96 \]
Now the equation becomes:
\[ 18x = 267.96 \]
Next, divide both sides by \( 18 \):
\[ x = \frac{267.96}{18} \]
Calculating \( \frac{267.96}{18} \):
\[ x \approx 14.88 \]
Thus, the length of \( x \) in decimal form to the nearest hundredth is:
\[ \boxed{14.88} \text{ inches} \]