#### Answer

{$-1$}

#### Work Step by Step

First, we add the two fractions on the left hand side of the equation by finding the LCM of the denominators of the two fractions.
Upon observation, we find that the LCM of the two denominators is 36. Then, we cross multiply the resultant fractional equation and solve the resultant expression.
Step 1: $\frac{t-3}{4}+\frac{t+1}{9}=-1$
Step 2: $\frac{9(t-3)+4(t+1)}{36}=-1$
Step 3: $\frac{9t-27+4t+4}{36}=-1$
Step 4: $\frac{9t+4t+4-27}{36}=-1$
Step 5: $\frac{13t-23}{36}=-1$
Step 6: $13t-23=-1(36)$
Step 7: $13t=-36+23$
Step 8: $13t=-13$
Step 9: $t=-1$
Therefore, the solution set is {$-1$}.