1. **State the problem:** Find the value of $x$ such that the line passing through the points $(-11, -11)$ and $(x, 1)$ has a slope of $3$. 2. **Formula for slope:** The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Apply the formula:** Substitute $m=3$, $(x_1, y_1) = (-11, -11)$, and $(x_2, y_2) = (x, 1)$: $$3 = \frac{1 - (-11)}{x - (-11)} = \frac{1 + 11}{x + 11} = \frac{12}{x + 11}$$ 4. **Solve for $x$:** Multiply both sides by $x + 11$: $$3(x + 11) = 12$$ 5. **Distribute:** $$3x + 33 = 12$$ 6. **Isolate $x$:** $$3x = 12 - 33$$ $$3x = -21$$ 7. **Divide both sides by 3:** $$x = \frac{\cancel{3}x}{\cancel{3}} = \frac{-21}{3}$$ $$x = -7$$ **Final answer:** $$\boxed{-7}$$