# Solving an inequality with many parameters [closed]

I need to solve the below inequality for x. I am very new to Mathematica. Below syntax does not give me any output. Please help.

Reduce[E^[(2 b - T) (x μ - λ)] <= β - (2 a β λ)/(x ν) + (2 a β μ)/ν, x]


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• First, syntax is wrong. You can't write E^[]. May be it was cut/paste error. Second, do you know any assumptions on the parameters? Assumptions can help find solutions. – Nasser Dec 4 '17 at 5:54

Reduce[Exp[(2 b - T) (x μ - λ)] <= β - (2 a β λ)/(x ν) + (2 a β μ)/ν, x, Reals]


or

Reduce[E^((2 b - T) (x μ - λ)) <= β - (2 a β λ)/(x ν) + (2 a β μ)/ν, x, Reals]


But Mathematica can't solve this inequality without more information concerning the parameters. When you evaluate either you will get

Reduce::nsmet: This system cannot be solved with the methods available to Reduce.

• Thank you all for the answers. I have below assumptions on parameters: a<0, 2b-T<0, x μ - λ>0, x>0, actually I need to derive x algebraically, as my theoretical result. Then I can apply appropriate numerical values to all others to find x in my simulations. So isn't there anyway that I can solve this inequality using above information and derive x algebraically? – Babara Peters Dec 4 '17 at 21:33

There is no algebraic solution for x with that inequation. But may be the folowing procedure helps you.

According to the given conditions, Exp[...] ranges between 0 and 1 and therefore the right side of the equation is greater zero.

So get condtions for the parameters to satisfy at least.

red = Reduce[
0 < β - (2 a β λ)/(x ν) + (2 a β \
μ)/ν && x μ - λ > 0 && x > 0 && a < 0, x, Reals] //
LogicalExpand


You get 16 different sets of parameter conditions, for example

red[[4]]= (-(ν/(2 μ)) == a &&
x > λ/μ && β > 0 && λ > 0 && μ >
0 && ν > 0)


Now insert definite values for the parameters to get exact condition for x

Manipulate[
Reduce[Rationalize[
Exp[tbT (x μ - λ)] <= β - (2 a β \
λ)/(x ν) + (2 a β μ)/ν && (x > λ/\
μ && β > 0 && λ > 0 && μ > 0 && ν > 0) /.
a -> -(ν/(2 μ)), 0], x] // N, {ν, 1, 5}, {β, 1,
6}, {λ, 1, 4}, {μ, 1, 6}, {tbT, -1, -5},
ContinuousAction -> False]