To solve the equation:

$-4x + 7 = -11$

we will isolate $x$ by following these steps:

1. Subtract 7 from both sides: $-4x = -11 - 7$ $-4x = -18$

2. Divide both sides by -4 to solve for $x$: $x = \frac{-18}{-4} = \frac{18}{4} = 4.5$

Thus, the answer is: $\boxed{\text{E. } 4.5}$

Would you like a further breakdown of the steps? Here are some additional questions that build on this type of problem:

1. How would the solution change if the equation were $-4x + 7 = -13$?
2. What is the solution if the equation is $-5x + 7 = -18$?
3. How can you check if a solution is correct in a linear equation?
4. If the equation were $4x - 7 = 11$, what would be the value of $x$?
5. How does changing the sign of $x$ in the equation affect the solution?

Tip: When solving linear equations, always perform the same operation on both sides to keep the equation balanced.