Question

determine the range of side ab.

Explanation:

Step1: Apply triangle - inequality theorem in $\triangle ABC$

In $\triangle ABC$, by the triangle - inequality theorem, the difference between two sides is less than the third side and the sum of two sides is greater than the third side. So, $|9 - 5|\lt x\lt9 + 5$, which simplifies to $4\lt x\lt14$.

Step2: Apply triangle - inequality theorem in $\triangle ABD$

In $\triangle ABD$, we have $|22-17|\lt x\lt22 + 17$, which simplifies to $5\lt x\lt39$.

Step3: Find the intersection of the two ranges

To satisfy both triangle - inequality conditions for the two triangles sharing side $AB$, we take the intersection of the two ranges. The intersection of $4\lt x\lt14$ and $5\lt x\lt39$ is $5\lt x\lt14$.

Answer:

$5\lt x\lt14$