# Solutions of relations on the complex numbers

Sketch the sets of points determined by the following relations:

a) Re(z + 1) = 0

b) Im(z - 2 i) > 6.

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#### Solution Preview

(a) Let z = x + iy, where x and y are real numbers (i.e., x = Re(z) and y = Im(z)).

Substituting x + iy for z (in z + 1), we obtain z + 1 = (x + iy) + 1.

Combining the two terms that consist of real numbers (namely, x and 1), we get (x + iy) + 1 = (x + 1) + iy.

Thus z + 1 = (x + 1) + iy, hence Re(z + 1) = x + 1.

To solve the equation Re(z + 1) = 0, we set x + 1 to 0 and solve ...

#### Solution Summary

Two relations (one equation and one inequality) are given on the complex numbers. Their solutions are derived, and detailed explanations of the solutions are provided.

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