The paper contains new sufficient conditions under which every solution x of delay differential inequalities of the forms \[ x^{(n)}(t)sgn x(t)\geq p(t)\prod^{m}_{i=1}| x(g_ i(t))|^{r_ i} \] and \[ x^{(n)}(t)sgn x(t)\geq \sum^{m}_{i=1}p_ i(t)| x(g_ i(t))|, \] is either oscillatory or \(\lim_{t\to \infty} | x^{(k)}(t)| =\infty\) \((k=0,1,...,n-1)\) monotonically, where \(n\geq 3\), \(r_ i(i=1,...,m)\) are nonnegative numbers with \(r_ 1+...+r_ m=1\), the functions \(p,p_ i: {\mathbb{R}}_+\to {\mathbb{R}}_+=[0,\infty)\) are integrable on each finite segment and are not identically equal zero in every neighbourhood of infinity, the delay arguments \(g_ i: {\mathbb{R}}_+\to {\mathbb{R}}_+\) \((i=1,...,m)\) are nondecreasing functions with \(g_ i(t)\leq t\) on \({\mathbb{R}}_+\) and \(\lim_{t\to \infty} g_ i(t)=\infty.\)