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These are questions asked by toandhsp

In geometry 3D, I have four vertices A(x1,y1,z1), B(x2,y2,z2), C(x3,y3,z3) and D(x4,y4,z4) of a tetrahedron. How to check a point P(x0,y0,z0) inside, outside or lie on  tetrahedron ABCD?

This is a problem of Russian in 2005: Solve inequality 


The problem can solve by hand. We only solve

(3-x-sqrt(5-x^2))/((2*x-7)/4-(x-5)/4) >=0.

I tried



I got

RealRange(1,Open(2)), RealRange(Open(2),5^(1/2)).

If I solve directly by



I don't get the exact answer. 

P.S. with Mathematica, I only use

Reduce[(3 - x - Sqrt[5 - x^2])/(Cos[(2 x - 7)/4] - Cos[(x - 5)/4]) >=
0, x, Reals]



With Maple, I tried


And I got

Warning, solutions may have been lost


With Mathematica

Reduce[Sin[(x + 1) \[Pi]/(4 x^2 - 4 x + 2)] == Cos[(x - 2) \[Pi]/(4 x^2 - 4 x + 2)], x, Reals]

I got all solutions.

I tried


and got 

Warning, solutions may have been lost

RealRange(-infinity,Open(-5/2)), RealRange(Open(-5/2),Open(-2)), RealRange(Open(1),Open(3))

How can I get the exact the solution?

I want to solve the inequality sqrt(8*16^x - 1/2* 9^x) <= 3*4^x - 3^x

I tried 


solve(sqrt(8*16^x - 1/2* 9^x) <= 3*4^x - 3^x,x);

And I got 

Error, (in evala/inv/heuristic) cannot allocate memory (size=117506048).

How to solve the above inequality?

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