https://file.aiquickdraw.com/mathbot/file/1726598723426vl88rkme.jpg

The inequality provided in the image is:

\[
3(x - 5) + \frac{x - 2}{6} + \frac{4 - x}{3} < \frac{x + 8}{2}
\]

### Steps to solve the inequality:

1. **Distribute the terms**: Expand the expression \(3(x - 5)\).

\[
3(x - 5) = 3x - 15
\]

The inequality becomes:

\[
3x - 15 + \frac{x - 2}{6} + \frac{4 - x}{3} < \frac{x + 8}{2}
\]

2. **Combine like terms**: The fractions need a common denominator. The terms are \(\frac{x - 2}{6}\) and \(\frac{4 - x}{3}\). To simplify, rewrite \(\frac{4 - x}{3}\) as \(\frac{2(4 - x)}{6}\):

\[
3x - 15 + \frac{x - 2}{6} + \frac{2(4 - x)}{6} < \frac{x + 8}{2}
\]

Now, combine the terms inside the fractions:

\[
\frac{(x - 2) + 2(4 - x)}{6} = \frac{x - 2 + 8 - 2x}{6} = \frac{-x + 6}{6}
\]

The inequality becomes:

\[
3x - 15 + \frac{-x + 6}{6} < \frac{x + 8}{2}
\]

3. **Multiply both sides to eliminate the fractions**: Multiply the entire inequality by 6 to clear the denominators.

\[
6(3x - 15) + (-x + 6) < 3(x + 8)
\]

This simplifies to:

\[
18x - 90 - x + 6 < 3x + 24
\]

4. **Simplify the inequality**: Combine like terms on both sides.

\[
17x - 84 < 3x + 24
\]

5. **Move all terms involving \(x\) to one side and constants to the other**:

\[
17x - 3x < 24 + 84
\]

\[
14x < 108
\]

6. **Solve for \(x\)**:

\[
x < \frac{108}{14} = 7.71
\]

So, \(x < 7.71\).

### Conclusion:
The solution to the inequality is \(x < 7.71\).

Since the question asks for the largest integer that satisfies the inequality, the largest integer less than 7.71 is **7**.

Would you like a more detailed explanation or have any other questions? Here are some related questions:

1. How do you handle inequalities when there are variables on both sides?
2. What happens if the inequality involves negative coefficients?
3. How can you verify the solution of an inequality?
4. Why do we multiply both sides by the least common denominator in inequalities?
5. What changes if we multiply or divide an inequality by a negative number?

**Tip**: Always check your solution by plugging it back into the original inequality!

Resolver la siguiente inequación: 3(x - 5) + (x - 2)/6 + (4 - x)/3 < (x + 8)/2

Learn how to solve the rational inequality 3(x - 5) + (x - 2)/6 + (4 - x)/3 < (x + 8)/2 with step-by-step instructions. This problem involves algebraic concepts such as rational expressions and inequalities, suitable for students in Grades 9-10.

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