Question

If I x + 1 I ≤ 9 and I y - 1 I ≤ 7 then a ≤ 3y - 2x ≤ b . What is the value of b - a ?

Answer 1

| X + 1 |  <= 9   
        X + 1 <=  9   =>    X <= 8
        - (X + 1)  <=  9     =>  X >=  -10

| Y - 1 |  <= 7 
      Y+1 <= 7  and   -Y+1 <= 7    =>    - 6 <= Y <= 8

So draw coordinate axes and draw two lines at X = -10, and 8.
Then draw  two lines at Y = -6 and 8.
You get a rectangle, with corners A(-10, 8), B(8, 8),  C(8, -6) and D(-10, -6).

Expression :  let    3 Y - 2 X = K    then    Y = 2X/3 + K/3
This is a line with positive slope. By varying K, we can place this line.    It will enter the rectangular region ABCD at A(-10,8) and as you move the straight line to the right, it will leave the ABCD at C (8, -6).

      Hence  at A,   K = 3 * 8 - 2 *(-10) = 44
                 at  C,   K = 3 *(-6) - 2 * 8 = - 34

           So,     -34 <= 3Y - 2X <= 44 
       a = -34     and    b = 44 

        Hence,   b - a =  78

You can draw a diagram with coordinate axes and see. Then it will be easier to understand.