No solution with "Solve"

Questions
,
1

Hey guys,
have a look at the following:

In[1]:= Solve[x^2 + 4 > 5*Abs[x], x]

During evaluation of In[1]:= Solve::fulldim: The solution set contains a full-dimensional component; use Reduce for complete solution information.

Out[1]= {{}}

As you can see, Wolfram Mathematica is not able to provide a correct solution.
But the equation should definitely have a solution, I tried it with Maple.

Can you guys help me out with this problem?

Greetings.

2

Did you try what is suggested in the warning message? It seems like fairly clear wording.

3

Hey, to be honest I do not know how to use “Reduce” now.
Should I apply it like that?

In[14]:= Reduce[Solve[x^2 + 4 > 5*Abs[x], x]]

During evaluation of In[14]:= Solve::fulldim: The solution set contains a full-dimensional component; use Reduce for complete solution information.

During evaluation of In[14]:= Reduce::naqs: {} is not a quantified system of equations and inequalities.

Out[14]= Reduce[{{}}]

That does not help.

Can you help me out?

Greetings.

4

Check Help > Wolfram Documentation > Reduce for details on how to use that function.

5

Hey, thank you for answering:

In[15]:= Reduce[x^2 + 4 > 5*Abs[x], x]

Out[15]= x < -4 || -1 < x < 1 || x > 4

This actually works. Nice.
Can you tell me why?

The solution set contains a full-dimensional component I don’t really know what that should mean.

Greetings.

6

Solve gives you a set of solutions of the form of replacement rules {{x->s1}, {s->s2},...} where s1 et al. are the solutions. Reduce gives you a reduced form for the input expression.

How would you expect Solve to express this solution set as replacement rules? I suppose it could yield a discontinuous Piecewise function of a real parameter, but I think that’s obscure and unlikely to be useful. So, it gives you an error, obscure as to the cause, but clear and concrete about the fix. While the solution set isn’t reasonably represented by replacement rules, reducing an expression to an equivalent expression is possible here, and seems a useful way to represent the solution set.

7

Ah, understood.

Thank you.

Greetings.