Motorized Bicycles in Texas can be defined as "motor assisted scooters" and not "mopeds". When you take this translation of Texas State Code 551.301, you see that motorized bikes are legal up to 40cc and able to operate on roads w/ speed limits posted at 35 mph. I guess this means you can drive at 35 mph on smaller side streets, but not big highways?
"Motor assisted scooter":
(A) means a self-propelled device with:
(i) at least two wheels in contact with the ground during operation;
(ii) a braking system capable of stopping the device under typical operating conditions;
(iii) a gas or electric motor not exceeding 40 cubic centimeters;
(iv) a deck designed to allow a person to stand or sit while operating the device; and
(v) the ability to be propelled by human power alone
Read Subchapter E-5 on this page
http://tlo2.tlc.state.tx.us/statutes/docs/TN/content/pdf/tn.007.00.000551.00.pdf