We treat systems of real diagonal forms
F
1
(
x
)
,
F
2
(
x
)
,
…
,
F
R
(
x
)
F_1(\mathbf {x}), F_2(\mathbf {x}), \ldots , F_R(\mathbf {x})
of degree
k
k
, in
s
s
variables. We give a lower bound
s
0
(
R
,
k
)
s_0(R,k)
, which depends only on
R
R
and
k
k
, such that if
s
≥
s
0
(
R
,
k
)
s \geq s_0(R,k)
holds, then, under certain conditions on the forms, and for any positive real number
ϵ
\epsilon
, there is a nonzero integral simultaneous solution
x
∈
Z
s
\mathbf {x} \in \mathbb {Z}^s
of the system of Diophantine inequalities
|
F
i
(
x
)
|
>
ϵ
|F_i(\mathbf {x})| > \epsilon
for
1
≤
i
≤
R
1 \leq i \leq R
. In particular, our result is one of the first to treat systems of inequalities of even degree. The result is an extension of earlier work by the author on quadratic forms. Also, a restriction in that work is removed, which enables us to now treat combined systems of Diophantine equations and inequalities.