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As noted previously above, other rules apply if the charges stemmed from family violence or other crimes against persons..
Note that time is determined from date conviction is entered, not date of offense.

IANAL, My opinion. Not legal advice.

GC §411.172. ELIGIBILITY.
Text of subsection effective on Jan. 1, 2016
(a) A person is eligible for a license to carry a handgun if the person:
...............
(8) has not, in the five years preceding the date of application, been convicted of a Class A or Class B misdemeanor or equivalent offense or of an offense under Section 42.01, Penal Code, or equivalent offense;
(9) is fully qualified under applicable federal and state law to purchase a handgun;