Any way to solve parametric inequations?

[Albert Chan]

(09-14-2021 05:13 AM)jte Wrote:  

(09-13-2021 07:42 PM)Albert Chan Wrote:  ⋮
CAS> assume(y≥0)
CAS> solve(x*y/(x+y)^2 ≤ 0, x)       → {x<(-y),(x>(-y)) AND (x≤0)}
⋮

(It seems to me that the x>0,y=0 portion is missing.)

Good catch ! Another bug report ...

Perhaps it should be setup with 2 assumes (y≥0, y≤0), like ti-89 solution.
Or, "safer" assumes (y>0, y<0), and handle y==0 edge case separately.
Better yet, implement CAS do assume stuff automatically !

(09-13-2021 07:42 PM)Albert Chan Wrote:  Or, for y≠0, reduce to simple inequality, with t = x/y

CAS> solve(t/(t+1)^2 ≤ 0, t)            → {t<-1,(t>-1) AND (t≤0)}

Translate t = x/y back to (x,y) world ...
First, edge case out of the way: if (y==0) then return (x!=0)

(SIGN(x)!=SIGN(y)) AND (x!=-y)

With SIGN(), we could remove test for edge case
Unfortunately, CAS SIGN() have bugs (HOME side, SIGN() seems OK)

CAS> SIGN([-1E-10, 0., 1E-10]       → [-1., 0., -1.]
CAS> SIGN([-1E-12, 0., 1E-12])       → [0., 0., 0.]
CAS> SIGN(+inf)          → 1
CAS> SIGN(1E400)       → undef