The inequality in the image is:

$\frac{8}{3} - \frac{1}{2}x \geq \frac{5}{6} + x$

Let's solve this step-by-step:

Step 1: Eliminate fractions by finding a common denominator

The least common denominator (LCD) of 3, 2, and 6 is 6. Multiply the entire inequality by 6 to eliminate the fractions:

$6 \times \left(\frac{8}{3} - \frac{1}{2}x \right) \geq 6 \times \left( \frac{5}{6} + x \right)$

This simplifies to:

$6 \times \frac{8}{3} - 6 \times \frac{1}{2}x \geq 6 \times \frac{5}{6} + 6x$

$16 - 3x \geq 5 + 6x$

Step 2: Combine like terms

Move the terms involving $x$ to one side and constants to the other side:

$16 - 5 \geq 6x + 3x$

Simplify:

$11 \geq 9x$

Step 3: Solve for $x$

Now, solve for $x$ by dividing both sides by 9:

$\frac{11}{9} \geq x$

This can also be written as:

$x \leq \frac{11}{9}$

Thus, the solution to the inequality is:

$x \leq \frac{11}{9}$

Do you need further details or explanations?

Here are 5 related questions:

1. How do we solve inequalities that involve multiple variables?
2. What are the steps to solving linear equations?
3. How does multiplying by the least common denominator help eliminate fractions?
4. Can inequalities be solved using the same method as equations?
5. What happens if you divide or multiply both sides of an inequality by a negative number?

Tip: When solving inequalities, always remember that multiplying or dividing by a negative number flips the inequality sign!