To find the length of \( x \) in the big triangle, we can use the concept of similar triangles. The triangles are similar because corresponding angles are equal, and the ratios of the lengths of corresponding sides are equal.
Let \( s_1 \) be a side of the small triangle, \( s_2 \) be the corresponding side of the big triangle, and \( k \) be the scale factor between the two triangles. The relationship between the sides of the triangles can be expressed as:
\[ \frac{s_1}{s_2} = \frac{s_1'}{s_2'} \]
where \( s_1 = 11.4 \) inches (small triangle), \( s_1' = x \) (big triangle), \( s_2 = 18 \) inches (small triangle), and \( s_2' = 23.4 \) inches (big triangle).
Thus, we can set up the proportion:
\[ \frac{11.4}{x} = \frac{18}{23.4} \]
Now, we can solve for \( x \). First, rearranging the equation gives:
\[ 11.4 \cdot 23.4 = 18 \cdot x \]
Now we calculate \( 11.4 \cdot 23.4 \):
\[ 11.4 \times 23.4 = 267.96 \]
Now substitute back into the equation:
\[ 267.96 = 18x \]
Next, we solve for \( x \) by dividing both sides by 18:
\[ x = \frac{267.96}{18} \approx 14.88 \]
Therefore, the length of \( x \) to the nearest hundredth is:
\[ \boxed{14.88} \text{ inches} \]