  In this paper we prove an asymptotic formula for the number of solutions in
prime numbers to systems of simultaneous linear inequalities with algebraic
coefficients. For $m$ simultaneous inequalities we require at least $m+2$
variables, improving upon existing methods, which generically require at least
$2m+1$ variables. Our result also generalises the theorem of Green-Tao-Ziegler
on linear equations in primes. Many of the methods presented apply for
arbitrary coefficients, not just for algebraic coefficients, and we formulate a
conjecture concerning the pseudorandomness of sieve weights which, if resolved,
would remove the algebraicity assumption entirely.
