a) To solve the equation |x + 2| - x = 2, we can split it into two cases based on the sign of x + 2.

Case 1: x + 2 ≥ 0
In this case, the absolute value |x + 2| is equal to x + 2. Substituting this into the equation, we get (x + 2) - x = 2. Simplifying, we have 2 = 2, which is always true. Therefore, any value of x that is greater than or equal to -2 satisfies the equation in this case.

Case 2: x + 2 < 0
In this case, the absolute value |x + 2| is equal to -(x + 2). Substituting this into the equation, we get -(x + 2) - x = 2. Simplifying, we have -2x - 2 = 2. Adding 2 to both sides, we get -2x = 4. Dividing both sides by -2, we have x = -2. Therefore, x = -2 is the solution in this case.

Combining the solutions from both cases, the solution to the equation |x + 2| - x = 2 is x = -2 and x ≥ -2.

b) To solve the equation |x - 3| = 21, we can split it into two cases based on the sign of x - 3.

Case 1: x - 3 ≥ 0
In this case, the absolute value |x - 3| is equal to x - 3. Substituting this into the equation, we get x - 3 = 21. Adding 3 to both sides, we have x = 24. Therefore, x = 24 is the solution in this case.

Case 2: x - 3 < 0
In this case, the absolute value |x - 3| is equal to -(x - 3). Substituting this into the equation, we get -(x - 3) = 21. Simplifying, we have -x + 3 = 21. Subtracting 3 from both sides, we get -x = 18. Dividing both sides by -1, we have x = -18. Therefore, x = -18 is the solution in this case.

Combining the solutions from both cases, the solution to the equation |x - 3| = 21 is x = -18 and x = 24.

c) To solve the equation |2x + 3| - |x - 3| = 0, we can split it into two cases based on the sign of 2x + 3 and x - 3.

Case 1: 2x + 3 ≥ 0 and x - 3 ≥ 0
In this case, the absolute values |2x + 3| and |x - 3| are equal to 2x + 3 and x - 3, respectively. Substituting these into the equation, we get (2x + 3) - (x - 3) = 0. Simplifying, we have x + 6 = 0. Subtracting 6 from both sides, we get x = -6. Therefore, x = -6 is the solution in this case.

Case 2: 2x + 3 < 0 and x - 3 < 0
In this case, the absolute values |2x + 3| and |x - 3| are equal to -(2x + 3) and -(x - 3), respectively. Substituting these into the equation, we get -(2x + 3) - -(x - 3) = 0. Simplifying, we have -2x - 3 + x - 3 = 0. Combining like terms, we have -x - 6 = 0. Adding 6 to both sides, we get -x = 6. Dividing both sides by -1, we have x = -6. Therefore, x = -6 is the solution in this case.

Combining the solutions from both cases, the solution to the equation |2x + 3| - |x - 3| = 0 is x = -6.