Solving the Inequality $2 – 5x \le \frac{6x-5}{3}$

To find the values of $x$ satisfying the inequality, follow these steps:

1. Start with the given inequality:

   $2 – 5x \le \frac{6x-5}{3}$

2. Multiply both sides by 3 to clear the fraction. Since 3 is positive, the inequality direction remains unchanged:

   $3(2 – 5x) \le 6x-5$

3. Distribute the 3 on the left side:

   $6 – 15x \le 6x-5$

4. Gather the $x$ terms on one side and the constant terms on the other. Add $15x$ to both sides:

   $6 \le 6x + 15x - 5$

   $6 \le 21x - 5$

5. Add 5 to both sides:

   $6 + 5 \le 21x$

   $11 \le 21x$

6. Divide both sides by 21. Since 21 is positive, the inequality direction stays the same:

   $\frac{11}{21} \le x$

7. Rewrite the inequality in the standard form:

   $x \ge \frac{11}{21}$

Number Line Representation

The solution $x \ge \frac{11}{21}$ means $x$ can be any real number greater than or equal to $\frac{11}{21}$.

• On a number line, this is represented by a closed circle (or a bold dot) at $\frac{11}{21}$, indicating that this value is included in the solution set.
• The line segment extends to the right from $\frac{11}{21}$ towards positive infinity, showing all values greater than $\frac{11}{21}$ are included.

This representation matches the one shown in Option 3.