To find the value of \( x \) in the big triangle, we can use the concept of similar triangles. The ratios of the corresponding sides of similar triangles are equal.
Given:
- Side of the small triangle: 11.4 inches (corresponds to \( x \))
- Side of the small triangle: 18 inches (corresponds to 23.4 inches in the big triangle)
We can set up the ratio as follows:
\[ \frac{x}{11.4} = \frac{23.4}{18} \]
Now, we can cross-multiply to solve for \( x \):
\[ x \cdot 18 = 11.4 \cdot 23.4 \]
Calculating \( 11.4 \cdot 23.4 \):
\[ 11.4 \cdot 23.4 = 267.96 \]
Now substituting back into the equation:
\[ 18x = 267.96 \]
Now, divide both sides by 18 to solve for \( x \):
\[ x = \frac{267.96}{18} \]
Calculating that gives:
\[ x = 14.88 \]
Thus, the length of \( x \) in decimal form to the nearest hundredth is:
\[ \boxed{14.88} \text{ inches} \]