Question on Formal Definition of Limits

A machinist must do all of his work to within a tolerance of 10^-3 mm. The calibrations on his machine are such that if the machinist's settings are accurate to within m mm, then the dimensions of the product have a tolerance of 1.2|m| + 100m^2. What accuracy of the settings is required to produce work of the desired tolerance.

The solutions manual says that using the equation 1.2|m| + 100m^2 < 10^-3 will yield the answer, but don't you have to use the rigorous definition of limits, which would give the equation
|1.2|m| +100m^2 - L| < 10^-3. This yields a different answer.

Which is the correct method?