As z is a complex number let it be x + yi. z' = x - yi.

If z/2 + 3=z'/3 - 2

=> (x + yi)/2 + 3 = (x - yi)/3 - 2

=> (x + yi)/2 - (x -yi)/3 = -5

=> x/2 - x/3 - i( y/2 - y/3) = -5

=> x/2 - x/3 = -5

=> (3x - 2x)/6 = -5

=> x = -30

y = 0 as there is no imaginary component on the opposite side.

Therefore **z = -30**

We'll write the rectangular form of the complex number z:

z = a + bi

z' is the conjugate of z:

z' = a - bi

To determine z, we'll have to determine it's coefficients:

(a + bi)/2 + 3 = (a - bi)/3 - 2

We'll multiply by 6 both sides:

3(a + bi) + 18 = 2(a - bi) - 12

We'll remove the brackets:

3a + 3bi + 18 = 2a - 2bi - 12

We'll move all terms to the left side:

3a - 2a + 3bi + 2bi + 18 + 12 = 0

a + 5bi + 30 = 0

The real part of the complex number from the left side is:

Re(z) = a + 30

The real part of the complex number from the right side is:

Re(z) = 0

Comparing, we'll get:

a + 30 = 0

a = -30

The imaginary part of the complex number from the left side is:

Im(z) = 5b

The imaginary part of the complex number from the right side is:

Im(z) = 5b

Comparing, we'll get:

5b = 0

b = 0

**The complex number z is: ****z = -30 + 0*i**