# Solving inequality using Reduce\[ \]?

**URL:** <https://community.wolfram.com/t/solving-inequality-using-reduce/17098>\
**Category:** Questions\
**Tags:** equation-solving\
**Created:** [November 25, 2020, 6:14pm UTC](https://community.wolfram.com/t/solving-inequality-using-reduce/17098 "2020-11-25T18:14:52Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![fedorrakitin](https://avatars.discourse-cdn.com/v4/letter/f/59ef9b/32.png) [@fedorrakitin](https://community.wolfram.com/u/fedorrakitin)\
**Post date:** [November 25, 2020, 6:14pm UTC](https://community.wolfram.com/t/solving-inequality-using-reduce/17098/1 "2020-11-25T18:14:52Z")

</div>

I’ve no idea how to solve in equation in Integers

```
Reduce[Log[2, 3 x + 5 y - 6 z + 3] > z^2 - 9 z + 16, {x,y,z}, Integers]

```

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**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\
**Post date:** [November 26, 2020, 10:14am UTC](https://community.wolfram.com/t/solving-inequality-using-reduce/17098/2 "2020-11-26T10:14:34Z")

</div>

The solution in reals is simple enough:

```
Reduce[Log[2, 3 x + 5 y - 6 z + 3] > z^2 - 9 z + 16,
  y, Reals] // Simplify

y > 1/5 (-3 + 2^(16 - 9 z + z^2) - 3 x + 6 z)

```

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<div class="post-metadata">

**Author:** ![fedorrakitin](https://avatars.discourse-cdn.com/v4/letter/f/59ef9b/32.png) [@fedorrakitin](https://community.wolfram.com/u/fedorrakitin)\
**Post date:** [November 26, 2020, 12:12pm UTC](https://community.wolfram.com/t/solving-inequality-using-reduce/17098/3 "2020-11-26T12:12:32Z")

</div>

Thx for answer, but there are 3 variables and I have to find integer solutions

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<div class="post-metadata">

**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\
**Post date:** [November 26, 2020, 1:45pm UTC](https://community.wolfram.com/t/solving-inequality-using-reduce/17098/4 "2020-11-26T13:45:31Z")

</div>

Take arbitrary integers `x,z` and you get an easy inequality for integer `y`.

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<div class="post-metadata">

**Author:** ![hdolhaine](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/hdolhaine/32/35924_2.png) [@hdolhaine](https://community.wolfram.com/u/hdolhaine)\
**Post date:** [November 26, 2020, 1:49pm UTC](https://community.wolfram.com/t/solving-inequality-using-reduce/17098/5 "2020-11-26T13:49:47Z")

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At least there are solutions, found by trial and error and given as { x, y, z }:

```
sol = {{-3, 5, 2}, {-3, 5, 3}, {-2, 4, 2}, {-2, 5, 2}, {-2, 5, 
   3}, {-1, 4, 2}, {-1, 4, 3}, {-1, 5, 2}, {-1, 5, 3}, {-1, 5, 4}, {0,
    3, 2}, {0, 4, 2}, {0, 4, 3}, {0, 5, 2}, {0, 5, 3}, {0, 5, 4}, {1, 
   3, 2}, {1, 3, 3}, {1, 4, 2}, {1, 4, 3}, {1, 4, 4}, {1, 5, 2}, {1, 
   5, 3}, {1, 5, 4}, {1, 5, 5}, {2, 2, 2}, {2, 2, 3}, {2, 3, 2}, {2, 
   3, 3}, {2, 4, 2}, {2, 4, 3}, {2, 4, 4}, {2, 5, 2}, {2, 5, 3}, {2, 
   5, 4}, {2, 5, 5}, {3, 1, 2}, {3, 2, 2}, {3, 2, 3}, {3, 3, 2}, {3, 
   3, 3}, {3, 3, 4}, {3, 4, 2}, {3, 4, 3}, {3, 4, 4}, {3, 4, 5}, {3, 
   5, 2}, {3, 5, 3}, {3, 5, 4}, {3, 5, 5}, {4, 1, 2}, {4, 1, 3}, {4, 
   2, 2}, {4, 2, 3}, {4, 2, 4}, {4, 3, 2}, {4, 3, 3}, {4, 3, 4}, {4, 
   4, 2}, {4, 4, 3}, {4, 4, 4}, {4, 4, 5}, {4, 5, 2}, {4, 5, 3}, {4, 
   5, 4}, {4, 5, 5}, {5, 0, 2}, {5, 1, 2}, {5, 1, 3}, {5, 2, 2}, {5, 
   2, 3}, {5, 2, 4}, {5, 3, 2}, {5, 3, 3}, {5, 3, 4}, {5, 3, 5}, {5, 
   4, 2}, {5, 4, 3}, {5, 4, 4}, {5, 4, 5}, {5, 5, 2}, {5, 5, 3}, {5, 
   5, 4}, {5, 5, 5}}

(Log[2, 3 #1 + 5 #2 - 6 #3 + 3] > #3^2 - 9 #3 + 16) & @@@ sol

```