Is there a method for solving DIFFICULT inequalities using Mupad?

I want to solve the following inequality:

solve(E^((-PI*(0.975/sqrt(15.2(1+k))))/sqrt(1-0.975^2/(15.2(1+k)))) < 10 , k, Real)

Mupads solution is just

R

Answer 1

[Walter Roberson]

Provided that E^ means exp(), in R2016a,

syms k
eqn = exp(-3705*pi/((95+95*k)^(1/2)*(2310400-144495/(1+k))^(1/2)))
solve(eqn<10,k)

gives me the empty set, and solve(eqn==10,k) gives me a number slightly larger than 1, which is a false root. There are no real solutions to eqn==10

A different package tells me the solution is (-22799/24320,infinity] where that lower bound is excluded. That lower bound is the point at which eqn would equal 0, except that right at that bound you have a division by 0. All values more negative than that lead to complex values. The equation is true from that point onwards because the value of the equation never exceeds 1, and 1 < 10

I have not yet found a way to get the symbolic toolbox to return a non-empty solution

Answer 2

[John D'Errico]

Inequalities generally don't have a "solution". Usually, they have an entire locus of solutions. In one variable, thus a problem like this, those sets will be intervals. They may possibly be infinite intervals, open intervals, or closed intervals.

You can identify the points bracketing the intervals of interest by solving the problem as an equality. But first, you might consider plotting the relationship. In fact, if I assume that E is exp(1), then it appears that the expression as you have written it is always less than 10, for all k. That may be why Mupad suggests the solution set is R, thus -inf < k < inf.

[Walter Roberson]

For k more negative than the value I posted, the expression is complex valued. Complex values are not orderable against real so they cannot be said to be less than 10 so the interval should not be all of R.

To return R, you need to have asked the wrong question, namely set of k such that eqn1 at k is not 10 or greater. Complex values are not greater than 10 so all of R is in that set.

I just realized that I did not try running from within a MuPad notebook; running from the MATLAB prompt gets me the empty set and a warning in r2016a