To solve the equation
\[ \frac{q + 6}{5} + \frac{q - 2}{2} = \frac{8}{5}, \]
we first want to eliminate the fractions. We can do this by finding a common denominator. The least common multiple of 5 and 2 is 10.
Now, we will multiply both sides of the equation by 10:
\[ 10 \cdot \left(\frac{q + 6}{5}\right) + 10 \cdot \left(\frac{q - 2}{2}\right) = 10 \cdot \left(\frac{8}{5}\right). \]
This simplifies to
\[ 2(q + 6) + 5(q - 2) = 16. \]
Now, distribute on both sides:
\[ 2q + 12 + 5q - 10 = 16. \]
Combine like terms:
\[ (2q + 5q) + (12 - 10) = 16 \implies 7q + 2 = 16. \]
Next, we isolate \(q\) by subtracting 2 from both sides:
\[ 7q = 14. \]
Now, divide both sides by 7:
\[ q = 2. \]
Checking the Solution
Now we substitute \(q = 2\) back into the original equation to verify:
\[ \frac{2 + 6}{5} + \frac{2 - 2}{2} = \frac{8}{5}. \]
This simplifies to:
\[ \frac{8}{5} + \frac{0}{2} = \frac{8}{5} \implies \frac{8}{5} + 0 = \frac{8}{5}. \]
Since both sides are equal, the solution \(q = 2\) is correct.
Thus, the solution is
\[ \boxed{2}. \]