Classically, it is known that every algebraic variety is birational to a hypersurface in some projective or affine space. Using generic linear projections, Doherty proved that over the complex numbers, this hypersurface can be taken to have at worst semi-log canonical singularities in dimensions up to five. This result extends classically known results for curves and surfaces. We present a positive-characteristic analogue of Doherty's theorem by showing that the resulting hypersurfaces are F-pure. This proves some cases of a conjecture of Bombieri, Andreotti, and Holm. To prove our result, we study F-injective singularities, which are the analogue of Du Bois singularities. This work is joint with Rankeya Datta.