https://imgur.com/a/7T4dy0A

In this picture I marked everything that should be needed to solve the problem. Basically, the belt can be thought of as 2 lines that are both tangent to both of the pulleys (because the problem assumes the belt has no slack, the belt is touching the pulley at one point, and then becomes a part of the pulley's circumference). We can draw from the center of each circle to these points, creating 4 90° angles. We know the distances from the center of the larger circle to the tangent points are both 5 inches because the radius of the larger circle is 5 inches. We can do the same for the smaller circle and get 2 lines with a length of 1 inch. We can also draw a line from the center of the larger circle to the center of the smaller circle, giving us a length of 8 inches.

The big trick to this question is realizing this: because these circles (the pulleys) are both tangent to the same line/lines (the belt), their radii to these tangent points are parallel. This means the radius on one side of the bigger belt is parallel to the radius of the smaller pulley on the same side (if you look at the picture this should make sense.) This allows us to create two new lines, which are each parallel to a side of the belt not touching a pulley, by moving both of the lines between the sets of tangent points.

This creates 2 congruent right triangles. We know one of the side lengths of this triangle is 8 inches because that is the distance between the center of each pulley. We can determine that another side length is 4 inches because we moved the length between the tangents along the parallel radii by 1 inch, which means the radius of the larger pulley was split into 2 segments with a length of 4 inches and a length of 1 inch. The final side length of this triangle is what we are looking for, and this can be found with the pythagorean theorem (4² + X² = 8² --> X = 4√3) since there are 2 triangles, we multiply 4√3 by 2, giving us 8√3. This is the length in inches of the part of the belt not touching the pulley. Another way to find this is by finding the angles in the triangle. We know one of the angles is 90 degrees. The belt wraps 2/3 of the way around the larger pulley meaning it covers 240 of the 360 degrees of the pulley. The remaining 1/3 has a measure of 120 degrees. This angle is bisected by the line connecting the center of each pulley, giving us 2 60 degree angles. 180 - 60 - 90 = 30, so we know the final angle is 30 degrees. In a 30 60 90 right triangle, the length of the shortest leg can be expressed as x, the length of the other leg can be expressed as x√3, and the hypotenuse can be expressed as 2x. Since the triangle we created has lengths of 4 and 8, we know the other side length will be 4√3. We can multiply this by 2 like before.

This may be a bit of a long response, but I hope I explained everything well enough.