The inequality provided in the image is:

$\frac{1}{2}x + 3 \geq 5x + 2$

Let's solve this step by step:

1. Subtract $\frac{1}{2}x$ from both sides to eliminate $x$ on the left-hand side: $3 \geq 5x - \frac{1}{2}x + 2$

   Simplify the right side: $3 \geq \left(5 - \frac{1}{2}\right)x + 2$

   This simplifies to: $3 \geq \frac{9}{2}x + 2$

2. Subtract 2 from both sides to isolate the $x$-term: $3 - 2 \geq \frac{9}{2}x$ Simplifies to: $1 \geq \frac{9}{2}x$

3. Multiply both sides by $\frac{2}{9}$ to solve for $x$: $x \leq \frac{2}{9}$

Final Answer:

$x \leq \frac{2}{9}$

This is the solution to the inequality.

Would you like further details or clarification on any of these steps?

Here are 5 related questions you can explore:

1. How do you graph the solution to this inequality on a number line?
2. What happens to the inequality if you multiply or divide by a negative number?
3. Can you solve similar inequalities involving fractions and variables?
4. How would you approach solving a quadratic inequality?
5. How can this inequality be represented as a system of linear inequalities?

Tip: Always reverse the inequality symbol when multiplying or dividing both sides by a negative number in an inequality.