Solve. \[ |z+10|=|z-5| \]

To solve the equation \( |z+10|=|z-5| \), we can interpret this geometrically. The expression \( |z + 10| \) represents the distance from the complex number \( z \) to the point \(-10\) on the complex plane, while \( |z - 5| \) represents the distance from \( z \) to the point \( 5\). The equation states that these two distances are equal, which means that the point \( z \) lies on the perpendicular bisector of the segment that connects the points \(-10\) and \( 5\). The midpoint of the segment between \(-10\) and \(5\) can be calculated as follows: \[ \text{Midpoint} = \frac{-10 + 5}{2} = \frac{-5}{2} = -2.5. \] The distance between \(-10\) and \(5\) is: \[ |5 - (-10)| = |5 + 10| = |15| = 15. \] The perpendicular bisector of this segment will be a vertical line that passes through \(-2.5\) since the line segment is horizontal (along the real axis). In the context of complex numbers, this line can be expressed as: \[ \text{Re}(z) = -2.5. \] Thus, any complex number \( z \) that satisfies this condition can be written as: \[ z = -2.5 + yi, \quad \text{where } y \in \mathbb{R}. \] In conclusion, the solutions to the equation \( |z + 10| = |z - 5| \) are all complex numbers that have a real part of \(-2.5\).