If |4x−5|+|3−x|≤2|4x−5|+|3−x|≤2, where x is a real number and maximum value of x is represented in th form of a/b in its lowest form, find the value of a+b? Any better ways of solving

Case 1: x >= 3, the inequality becomes 4x - 5 + x - 3 <= 2 => 5x <= 10 => x <= 2 => No solution in this interval
Case 2: 5/4 <= x < 3: The inequality becomes 4x - 5 + 3 - x <= 2 => 3x - 2 <= 2 => 3x <= 4 => x <= 4/3. So, 5/4 <= x <= 4/3
Case 3: x < 5/4: The inequality becomes 5 - 4x + 3 - x <= 2 => 5x >= 6 => x >= 6/5 => 6/5 <= x < 5/4
So, the maximum value of x is 4/3 => a + b = 4 + 3 = 7