The inequality given is:

$4x + 3 \leq 2x + 9$

Let's solve this step-by-step:

1. Isolate the variable terms on one side:
   Subtract $2x$ from both sides:

   $4x - 2x + 3 \leq 9$

   Simplifies to:

   $2x + 3 \leq 9$

2. Isolate the constant term on the other side:
   Subtract 3 from both sides:

   $2x \leq 6$

3. Solve for $x$:
   Divide both sides by 2:

   $x \leq 3$

Thus, the solution interval for the inequality is:

$(-\infty, 3]$

Would you like more details or have any other questions?

Here are some related questions to explore:

1. How do you solve inequalities involving absolute values?
2. What is the difference between solving an equation and an inequality?
3. Can you explain how to graph the solution set of an inequality on a number line?
4. How do you solve a quadratic inequality?
5. What are the rules for flipping the inequality sign when multiplying or dividing by a negative number?

Tip: Always check the final solution by substituting a number from the interval back into the original inequality.