# Questions tagged [inequalities]

Questions about handling with inequalities with, e.g., Reduce, Solve, FindInstance etc., their application in defining Regions, and ways to incorporate them into other computations, e.g. integration.

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### Cannot simplify or reduce a region

I have a region specified by a rather short expression, and the same region by a longer one, and cannot seem to reduce the size of the longer one. A simple example is ...
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### How can I solve this inequality for every $n$ in Mathematica?

I am trying to find the solution to the inequality $t^{-1/n}<1$ when $n$ is an integer and $t$ is positive, however when using Reduce[t^(-1/n) >= 1, t] it ...
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### Inequalities in the context of stability analysis

I find for the study of the active control of a system with 2 dofs to study the stability conditions. For that purpose, i use the Routh criteria and look when all the coefficients of the first row ...
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### Verifying simple inequality involving FactorialPower function

It is obviously true that FactorialPower[k, m] <= k^m when k and m are both positive ...
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### Verifying that a functions attends values below something

After plotting the graph of $$f(x)=\frac{π}{3\sqrt{2 π}} (\frac{e^{\frac{-x^2}{18}}}{3} + \frac{e^{\frac{-(x - 5)^2}{2}}}{1} +\frac{e^{\frac{-(x - 15)^2}{8}}}{2} ) (1 + x^2)$$ I would like to ...
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### Inequality plot like Wolfram Alpha

I'd like to reproduce in Mathematica the output of this Wolfram Alpha query. Of course I can just use WolframAlpha but I'd like to manipulate the graphics (add ...
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### How would I plot the inequality lines together with the intersection plot of the inequalities?

For example, I've got the following code: Plot[{x >= 5, y >= 8, x + 2y <= 64, x + y <= 40}] This would plot: Now, this is fair enough. It shows me ...
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### Inequality with assumptions

I would like to find 2D region formed by $(a,c) \in \mathbb{R}^2$ such that the inequality $f(a,c,x,y) \geq 0$ holds for all $x,y \in [0,1]$. If I solve this inequality with reduce of the form ...
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### FindInstance for all real values satisfying a condition

I seek at least one instance of positive fractions $p,q,r$ where $p,q,r \in \mathbb{Q}^+$, for which $$\frac12 (a - 2 b + c)^2 \ge pa^2 + qb^2 + rc^2\quad \forall\ a,b,c \in \mathbb{R}.$$ I used this ...
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### Solving a recurrence relation with variable coefficients

I am trying to solve this recurrence inequality for $a_n$, hopefully in terms of $a_0，a_1，c$ and $x$, $$(1-x)a_{n+1} \le xa_n + (1+x)a_{n-1} + c\quad\forall\ n\ge1$$ What I tried: RSolve. But ...
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### Simplify (or Reduce) a system of inequalities intelligently

I have the following system of inequalities of 4 real variables 3 (a + b) + 4 (d + c) > 0 && 2 (a - d) + 3 (b - c) > 0 The system can be written as ...
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### Connectivity of semi-algebraic set

I am given the following semi-algebraic set:  \{ x= \begin{bmatrix} x_1,x_2,x_3,x_4,x_5 \end{bmatrix} ^T\in\mathbb{R}^5| -2 \leq (1 - x_2x_3)x_5 - 2x_4x_3x_1 \leq 1.5,\\ ~ -1 \leq x_1 \leq 30,...
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### How to represent in a 3D graph the set of solutions to this system of inequalities?

I am studying this piecewise function: ...
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### How to get solutions for this system of inequalities?

I am new to Mathematica and I am struggling finding the solution for the following problem. I am studying this piecewise function: ...
Consider a $5$-uple $(r,a,b,c,s)$ with $a,b,c\in\mathbb{Z},s\in\mathbb{Q}_+,r\in\mathbb{Z}_+$, denote $\frac{1}{2}(a^2-b^2-c^2)-\frac{s}{r}$ by $\Delta(\mathbf{v})$. Let $A$ be a positive real number....