# No solution with "Solve"

**URL:** <https://community.wolfram.com/t/no-solution-with-solve/17021>\
**Category:** Questions\
**Tags:** algebra, equation-solving\
**Created:** [November 13, 2020, 2:30pm UTC](https://community.wolfram.com/t/no-solution-with-solve/17021 "2020-11-13T14:30:38Z")\
**Posts on this page:** 7\
**Page:** 1

<div class="post-metadata">

**Author:** ![christophweichenhain](https://avatars.discourse-cdn.com/v4/letter/c/eb9ed0/32.png) [@christophweichenhain](https://community.wolfram.com/u/christophweichenhain)\
**Post date:** [November 13, 2020, 2:30pm UTC](https://community.wolfram.com/t/no-solution-with-solve/17021/1 "2020-11-13T14:30:38Z")

</div>

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.

---

<div class="post-metadata">

**Author:** ![danl](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/danl/32/41267_2.png) [@danl](https://community.wolfram.com/u/danl)\
**Post date:** [November 13, 2020, 4:12pm UTC](https://community.wolfram.com/t/no-solution-with-solve/17021/2 "2020-11-13T16:12:08Z")

</div>

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

---

<div class="post-metadata">

**Author:** ![christophweichenhain](https://avatars.discourse-cdn.com/v4/letter/c/eb9ed0/32.png) [@christophweichenhain](https://community.wolfram.com/u/christophweichenhain)\
**Post date:** [November 13, 2020, 5:30pm UTC](https://community.wolfram.com/t/no-solution-with-solve/17021/3 "2020-11-13T17:30:27Z")

</div>

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.

---

<div class="post-metadata">

**Author:** ![danl](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/danl/32/41267_2.png) [@danl](https://community.wolfram.com/u/danl)\
**Post date:** [November 13, 2020, 5:32pm UTC](https://community.wolfram.com/t/no-solution-with-solve/17021/4 "2020-11-13T17:32:11Z")

</div>

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

---

<div class="post-metadata">

**Author:** ![christophweichenhain](https://avatars.discourse-cdn.com/v4/letter/c/eb9ed0/32.png) [@christophweichenhain](https://community.wolfram.com/u/christophweichenhain)\
**Post date:** [November 13, 2020, 6:05pm UTC](https://community.wolfram.com/t/no-solution-with-solve/17021/5 "2020-11-13T18:05:49Z")

</div>

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.

---

<div class="post-metadata">

**Author:** ![jpd](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/jpd/32/34994_2.png) [@jpd](https://community.wolfram.com/u/jpd)\
**Post date:** [November 14, 2020, 5:04pm UTC](https://community.wolfram.com/t/no-solution-with-solve/17021/6 "2020-11-14T17:04:31Z")

</div>

`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.

---

<div class="post-metadata">

**Author:** ![christophweichenhain](https://avatars.discourse-cdn.com/v4/letter/c/eb9ed0/32.png) [@christophweichenhain](https://community.wolfram.com/u/christophweichenhain)\
**Post date:** [November 14, 2020, 5:37pm UTC](https://community.wolfram.com/t/no-solution-with-solve/17021/7 "2020-11-14T17:37:32Z")

</div>

Ah, understood.

Thank you.

Greetings.