To determine the length of the segment \( A'B' \) after a reflection over the x-axis and a rotation of \( 293^\circ \) clockwise applied to the segment \( AB \), we can follow these steps:
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Reflection Over the X-Axis: When a point \( (x, y) \) is reflected over the x-axis, its new coordinates become \( (x, -y) \). However, this transformation does not change the distance between points; thus, the length of segment \( AB \) remains \( 19 \) inches after reflection.
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Rotation: Next, we rotate the segment \( 293^\circ \) clockwise. Similar to the reflection, rotation does not change the length of the segment. The distance between any two points remains the same regardless of their orientation in space.
Therefore, after both transformations—reflection over the x-axis and rotation of \( 293^\circ \) clockwise—the distance of \( A'B' \) remains the same as that of \( AB \).
Thus, we conclude: \[ A'B' = 19 \text{ inches}. \]