To solve the rational inequality:

$\frac{x - 1}{3} + 3 > 0$

Step 1: Simplify the inequality

Start by isolating the fraction by subtracting 3 from both sides:

$\frac{x - 1}{3} > -3$

Step 2: Eliminate the denominator

Multiply both sides of the inequality by 3 to get rid of the denominator:

$x - 1 > -9$

Step 3: Solve for $x$

Add 1 to both sides:

$x > -8$

Final Answer:

The solution to the inequality is:

$x > -8$

This means that $x$ must be greater than $-8$.

Would you like further details or clarification? Here are some related questions:

1. How would the inequality change if the numerator was a quadratic expression?
2. What if the sign of the inequality was less than $0$ instead of greater than $0$?
3. How can rational inequalities with variable denominators be solved?
4. How do you solve rational inequalities involving absolute values?
5. How do you solve compound inequalities?

Tip: Always be cautious about dividing or multiplying both sides of an inequality by a negative number—it reverses the inequality sign!