There is an interval $[a, b]$ that is the solution to the inequality \[|3x-80|\le|2x-105|\]  Find $a + b$.
1. Start with the given inequality:
   \[
   |3x - 80| \leq |2x - 105|
   \]

2. Square both sides to eliminate the absolute values:
   \[
   (3x - 80)^2 \leq (2x - 105)^2
   \]

3. Expand both sides:
   \[
   (3x - 80)^2 = 9x^2 - 480x + 6400
   \]
   \[
   (2x - 105)^2 = 4x^2 - 420x + 11025
   \]

4. Set up the inequality:
   \[
   9x^2 - 480x + 6400 \leq 4x^2 - 420x + 11025
   \]

5. Move all terms to one side of the inequality:
   \[
   9x^2 - 480x + 6400 - 4x^2 + 420x - 11025 \leq 0
   \]

6. Combine like terms:
   \[
   5x^2 - 60x - 4625 \leq 0
   \]

7. Factor the quadratic expression:
   \[
   5(x^2 - 12x - 925) \leq 0
   \]

8. Solve the quadratic equation \(x^2 - 12x - 925 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
   \[
   x = \frac{12 \pm \sqrt{144 + 3700}}{2}
   \]
   \[
   x = \frac{12 \pm \sqrt{3844}}{2}
   \]
   \[
   x = \frac{12 \pm 62}{2}
   \]

9. Find the roots:
   \[
   x = \frac{12 + 62}{2} = 37
   \]
   \[
   x = \frac{12 - 62}{2} = -25
   \]

10. The inequality \(5(x^2 - 12x - 925) \leq 0\) holds for \(x\) in the interval \([-25, 37]\).

11. Therefore, \(a = -25\) and \(b = 37\).

12. Calculate \(a + b\):
   \[
   a + b = -25 + 37 = 12
   \]

The final answer is \(\boxed{12}\)