Let's solve the inequality step-by-step.

The inequality given is:

$x - 9 \leq 8x + 5$

Step 1: Isolate the variable on one side.

To move the terms involving $x$ to one side, we can subtract $x$ from both sides:

$-9 \leq 7x + 5$

Step 2: Simplify the constant terms.

Now, subtract 5 from both sides to isolate the term with $x$:

$-9 - 5 \leq 7x$

This simplifies to:

$-14 \leq 7x$

Step 3: Solve for $x$.

To isolate $x$, divide both sides by 7:

$x \geq \frac{-14}{7}$

This simplifies to:

$x \geq -2$

Step 4: Write the solution in interval notation.

Since $x \geq -2$, the solution set in interval notation is:

$[-2, \infty)$

Final Answer:

The solution set is $[ -2, \infty )$.

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:

1. How do you solve inequalities when dividing or multiplying by a negative number?
2. What changes when solving a strict inequality (like $<$ instead of $\leq$)?
3. How do you graph the solution of $x \geq -2$ on a number line?
4. Can you explain the difference between interval notation and set-builder notation?
5. How would the solution change if the inequality were $x - 9 > 8x + 5$?

Tip: Always reverse the inequality sign when multiplying or dividing both sides by a negative number!