To solve the inequality $3x + 11 \leq 8x - 3$, we will follow these steps:

1. Isolate the variable on one side: Start by subtracting $3x$ from both sides: $11 \leq 8x - 3x - 3$ Simplifying this gives: $11 \leq 5x - 3$

2. Eliminate the constant term: Next, add 3 to both sides to move the constant term: $11 + 3 \leq 5x$ Simplifying further: $14 \leq 5x$

3. Solve for $x$: Divide both sides by 5 to isolate $x$: $\frac{14}{5} \leq x$ This can be written as: $x \geq \frac{14}{5}$

4. Write the solution in interval notation: The inequality $x \geq \frac{14}{5}$ in interval notation is: $\left[ \frac{14}{5}, \infty \right)$

Thus, the solution is $x \geq \frac{14}{5}$ or $\left[ \frac{14}{5}, \infty \right)$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How would the solution change if the inequality sign was reversed?
2. What is the decimal form of $\frac{14}{5}$?
3. How do you solve a compound inequality involving $x$?
4. Can you graph the solution on a number line?
5. What are the steps to solve a quadratic inequality?

Tip: Always double-check your steps when solving inequalities, especially when dividing or multiplying by negative numbers, as it reverses the inequality sign.