Almost sharp oscillation and nonoscillation criteria are obtained for the first-order nonlinear delay differential equation x'(t) + p(t)Pi(j=1)(m)\x(t - tau(j))\(alphaj)sign[x(t - tau(1))] = 0, t greater than or equal to t(0), where Sigma(j=1)(m) alpha(j) > 1. Some applications are given. (C) 2001 Elsevier Science.