2
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I want to solve this inequality for x1 when a,b,d, E, and alpha
are positive parameters
I tried this code and I did not get the required answer

Reduce[{((a*alpha*x1)/(b + x1) - d*alpha)*((a*alpha*x1)/(b + x1) - (d+E)*alpha) <= 0,x1 > 0}, x1]

I will appreciate any help

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  • 1
    $\begingroup$ E is a reserved symbol in Mathematica. $\endgroup$
    – Syed
    Oct 2 at 8:59
  • 1
    $\begingroup$ Reduce[{((a*\[Alpha]*x1)/(b + x1) - d*\[Alpha])*((a*\[Alpha]*x1)/(b + x1) - (d + e)*\[Alpha]) <= 0, x1 > 0}, x1, PositiveReals] ? $\endgroup$
    – Syed
    Oct 2 at 9:03
  • 1
    $\begingroup$ May I suggest a more manual approach? Your inequality has the form P*Q <= 0. This means that either P>=0 and Q<=0, or alternatively, P<=0 and Q>=0. You will probably get a better understanding if you first run Solve[P==0,x1] and Solve[Q==0,x1], this will tell you for which values x1 the factors P and Q change signs. $\endgroup$
    – user293787
    Oct 2 at 9:12
  • $\begingroup$ thanks a lot for your help. I will try this approach $\endgroup$ Oct 3 at 7:09

2 Answers 2

2
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First, as already mention replace E by e. Then define {a,b,d,e,x1}>0. With these changes you will get an answer:

 Reduce[{((a*alpha*x1)/(b + x1) - 
      d*alpha)*((a*alpha*x1)/(b + x1) - (d + e)*alpha) <= 
   0, {a, b, d, e, x1} > 0}, x1]

enter image description here

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3
  • $\begingroup$ thanks a lot for your help. I need to know what is the meaning of this || $\endgroup$ Oct 3 at 7:08
  • $\begingroup$ The name for || is Or. It means different possibilities. On the other hand, && is And and means conditions that must be fulfilled simultaneously. $\endgroup$ Oct 3 at 7:37
  • $\begingroup$ thank you so much $\endgroup$ Oct 3 at 10:14
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A more general answer is obtained in 13.1 on Windows 10 as follows.

Reduce[{((a*α*x1)/(b + x1) -  d*α)*((a*α*x1)/(b + x1) 
- (d + e)*α) <= 0,  x1 > 0}, x1, Reals]

*(e < 0 && ((d < 
    0 && ((a < 
        d + e && ((α < 0 && 
           b > 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b > 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)))) || (a == 
        d + e && ((α < 
            0 && ((b == 0 && x1 > 0) || (b > 0 && 
               x1 >= -((b d)/(-a + d))))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b == 0 && x1 > 0) || (b > 0 && 
               x1 >= -((b d)/(-a + d))))))) || (d + e < a < 
        d && ((α < 
            0 && ((b < 0 && 
               x1 >= (-b d - b e)/(-a + d + e)) || (b == 0 && 
               x1 > 0) || (b > 0 && 
               x1 >= -((b d)/(-a + d))))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               x1 >= (-b d - b e)/(-a + d + e)) || (b == 0 && 
               x1 > 0) || (b > 0 && 
               x1 >= -((b d)/(-a + d))))))) || (a == 
        d && ((α < 
            0 && ((b < 0 && 
               x1 >= (-b d - b e)/(-a + d + e)) || (b == 0 && 
               x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               x1 >= (-b d - b e)/(-a + d + e)) || (b == 0 && 
               x1 > 0))))) || (d < a < 
        0 && ((α < 0 && 
           b < 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b < 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))))) || (a == 0 && α == 
        0 && ((b <= 
            0 && (0 < x1 < (-b d - b e)/(d + e) || 
             x1 > (-b d - b e)/(d + e))) || (b > 0 && 
           x1 > 0))) || (a > 
        0 && ((α < 0 && 
           b < 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b < 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)))))) || (d == 
    0 && ((a < 
        e && ((α < 0 && b > 0 && 
           0 < x1 <= -((b e)/(-a + e))) || (α == 
            
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b > 0 && 
           0 < x1 <= -((b e)/(-a + e))))) || (a == 
        e && ((α < 0 && b >= 0 && 
           x1 > 0) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b >= 0 && 
           x1 > 0))) || (e < a < 
        0 && ((α < 
            0 && ((b < 0 && x1 >= -((b e)/(-a + e))) || (b >= 0 &&
                x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && x1 >= -((b e)/(-a + e))) || (b >= 0 &&
                x1 > 0))))) || (a == 
        0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 && 
           x1 > 0))) || (a > 
        0 && ((α < 0 && b < 0 && 
           0 < x1 <= -((b e)/(-a + e))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b < 0 && 
           0 < x1 <= -((b e)/(-a + e))))))) || (0 < 
    d < -e && ((a < 
        d + e && ((α < 
            0 && ((b < 0 && 
               0 < x1 <= -((b d)/(-a + d))) || (b > 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + 
                e)))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               0 < x1 <= -((b d)/(-a + d))) || (b > 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + e)))))) || (a == 
        d + e && ((α < 
            0 && ((b < 0 && 
               0 < x1 <= -((b d)/(-a + d))) || (b >= 0 && 
               x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               0 < x1 <= -((b d)/(-a + d))) || (b >= 0 && 
               x1 > 0))))) || (d + e < a < 
        0 && ((α < 
            0 && ((b < 
                0 && (0 < x1 <= -((b d)/(-a + d)) || 
                x1 >= (-b d - b e)/(-a + d + e))) || (b >= 0 && 
               x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 
                0 && (0 < x1 <= -((b d)/(-a + d)) || 
                x1 >= (-b d - b e)/(-a + d + e))) || (b >= 0 && 
               x1 > 0))))) || (a == 
        0 && ((b <= 
            0 && (0 < x1 < (-b d - b e)/(d + e) || 
             x1 > (-b d - b e)/(d + e))) || (b > 0 && 
           x1 > 0))) || (0 < a < 
        d && ((α < 
            0 && ((b < 
                0 && (0 < x1 <= (-b d - b e)/(-a + d + e) || 
                x1 >= -((b d)/(-a + d)))) || (b >= 0 && 
               x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 
                0 && (0 < x1 <= (-b d - b e)/(-a + d + e) || 
                x1 >= -((b d)/(-a + d)))) || (b >= 0 && 
               x1 > 0))))) || (a == 
        d && ((α < 
            0 && ((b < 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + e)) || (b >= 0 && 
               x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + e)) || (b >= 0 && 
               x1 > 0))))) || (a > 
        d && ((α < 
            0 && ((b < 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + e)) || (b > 0 && 
               0 < x1 <= -((b d)/(-a + d))))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + e)) || (b > 0 && 
               0 < x1 <= -((
                b d)/(-a + d))))))))) || (d == -e && ((a < 
        0 && ((α < 0 && b < 0 && 
           0 < x1 <= -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b < 0 && 
           0 < x1 <= -((b d)/(-a + d))))) || (a == 
        0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 && 
           x1 > 0))) || (0 < a < 
        d && ((α < 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b >= 0 &&
                x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b >= 0 &&
                x1 > 0))))) || (a == 
        d && ((α < 0 && b >= 0 && 
           x1 > 0) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b >= 0 && 
           x1 > 0))) || (a > 
        d && ((α < 0 && b > 0 && 
           0 < x1 <= -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b > 0 && 
           0 < x1 <= -((b d)/(-a + d))))))) || (d > -e && ((a < 
        
        0 && ((α < 0 && 
           b < 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b < 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))))) || (a == 0 && α == 
        0 && ((b <= 
            0 && (0 < x1 < (-b d - b e)/(d + e) || 
             x1 > (-b d - b e)/(d + e))) || (b > 0 && 
           x1 > 0))) || (0 < a < 
        d + e && ((α < 0 && 
           b < 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b < 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)))) || (a == 
        d + e && ((α < 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b == 0 &&
                x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b == 0 &&
                x1 > 0))))) || (d + e < a < 
        d && ((α < 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b == 0 &&
                x1 > 0) || (b > 0 && 
               x1 >= (-b d - b e)/(-a + d + e)))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b == 0 &&
                x1 > 0) || (b > 0 && 
               x1 >= (-b d - b e)/(-a + d + e)))))) || (a == 
        d && ((α < 
            0 && ((b == 0 && x1 > 0) || (b > 0 && 
               x1 >= (-b d - b e)/(-a + d + e)))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b == 0 && x1 > 0) || (b > 0 && 
               x1 >= (-b d - b e)/(-a + d + e)))))) || (a > 
        d && ((α < 0 && 
           b > 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b > 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))))))))) || (e == 
0 && ((d < 
    0 && ((a < 
        d && ((α < 0 && b > 0 && 
           x1 == -((b d)/(-a + d))) || (α == 
            
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b > 0 && 
           x1 == -((b d)/(-a + d))))) || (a == 
        d && ((α < 0 && b == 0 && 
           x1 > 0) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b == 0 && 
           x1 > 0))) || (d < a < 
        0 && ((α < 0 && b < 0 && 
           x1 == -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b < 0 && 
           x1 == -((b d)/(-a + d))))) || (a == 0 && α == 
        0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 && 
           x1 > 0))) || (a > 
        0 && ((α < 0 && b < 0 && 
           x1 == -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b < 0 && 
           x1 == -((b d)/(-a + d))))))) || (d == 
    0 && ((a < 0 && α == 
        0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 && 
           x1 > 0))) || (a == 
        0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 && 
           x1 > 0))) || (a > 0 && α == 
        0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 && 
           x1 > 0))))) || (d > 
    0 && ((a < 
        0 && ((α < 0 && b < 0 && 
           x1 == -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b < 0 && 
           x1 == -((b d)/(-a + d))))) || (a == 0 && α == 
        0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 && 
           x1 > 0))) || (0 < a < 
        d && ((α < 0 && b < 0 && 
           x1 == -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b < 0 && 
           x1 == -((b d)/(-a + d))))) || (a == 
        d && ((α < 0 && b == 0 && 
           x1 > 0) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b == 0 && 
           x1 > 0))) || (a > 
        d && ((α < 0 && b > 0 && 
           x1 == -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b > 0 && 
           x1 == -((b d)/(-a + d))))))))) || (e > 
0 && ((d < -e && ((a < 
        d && ((α < 0 && 
           b > 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b > 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))))) || (a == 
        d && ((α < 
            0 && ((b == 0 && x1 > 0) || (b > 0 && 
               x1 >= (-b d - b e)/(-a + d + e)))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b == 0 && x1 > 0) || (b > 0 && 
               x1 >= (-b d - b e)/(-a + d + e)))))) || (d < a < 
        d + e && ((α < 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b == 0 &&
                x1 > 0) || (b > 0 && 
               x1 >= (-b d - b e)/(-a + d + e)))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b == 0 &&
                x1 > 0) || (b > 0 && 
               x1 >= (-b d - b e)/(-a + d + e)))))) || (a == 
        d + e && ((α < 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b == 0 &&
                x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b == 0 &&
                x1 > 0))))) || (d + e < a < 
        0 && ((α < 0 && 
           b < 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b < 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)))) || (a == 
        0 && α == 
        0 && ((b <= 
            0 && (0 < x1 < (-b d - b e)/(d + e) || 
             x1 > (-b d - b e)/(d + e))) || (b > 0 && 
           x1 > 0))) || (a > 
        0 && ((α < 0 && 
           b < 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b < 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))))))) || (d == -e && ((a < 
        
        d && ((α < 0 && b > 0 && 
           0 < x1 <= -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b > 0 && 
           0 < x1 <= -((b d)/(-a + d))))) || (a == 
        d && ((α < 0 && b >= 0 && 
           x1 > 0) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b >= 0 && 
           x1 > 0))) || (d < a < 
        0 && ((α < 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b >= 0 &&
                x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && x1 >= -((b d)/(-a + d))) || (b >= 0 &&
                x1 > 0))))) || (a == 
        0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 && 
           x1 > 0))) || (a > 
        0 && ((α < 0 && b < 0 && 
           0 < x1 <= -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b < 0 && 
           0 < x1 <= -((b d)/(-a + d))))))) || (-e < d < 
    0 && ((a < 
        d && ((α < 
            0 && ((b < 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + e)) || (b > 0 && 
               0 < x1 <= -((b d)/(-a + d))))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + e)) || (b > 0 && 
               0 < x1 <= -((b d)/(-a + d))))))) || (a == 
        d && ((α < 
            0 && ((b < 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + e)) || (b >= 0 && 
               x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + e)) || (b >= 0 && 
               x1 > 0))))) || (d < a < 
        0 && ((α < 
            0 && ((b < 
                0 && (0 < x1 <= (-b d - b e)/(-a + d + e) || 
                x1 >= -((b d)/(-a + d)))) || (b >= 0 && 
               x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 
                0 && (0 < x1 <= (-b d - b e)/(-a + d + e) || 
                x1 >= -((b d)/(-a + d)))) || (b >= 0 && 
               x1 > 0))))) || (a == 
        
        0 && ((b <= 
            0 && (0 < x1 < (-b d - b e)/(d + e) || 
             x1 > (-b d - b e)/(d + e))) || (b > 0 && 
           x1 > 0))) || (0 < a < 
        d + e && ((α < 
            0 && ((b < 
                0 && (0 < x1 <= -((b d)/(-a + d)) || 
                x1 >= (-b d - b e)/(-a + d + e))) || (b >= 0 && 
               x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 
                0 && (0 < x1 <= -((b d)/(-a + d)) || 
                x1 >= (-b d - b e)/(-a + d + e))) || (b >= 0 && 
               x1 > 0))))) || (a == 
        d + e && ((α < 
            0 && ((b < 0 && 
               0 < x1 <= -((b d)/(-a + d))) || (b >= 0 && 
               x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               0 < x1 <= -((b d)/(-a + d))) || (b >= 0 && 
               x1 > 0))))) || (a > 
        d + e && ((α < 
            0 && ((b < 0 && 
               0 < x1 <= -((b d)/(-a + d))) || (b > 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + 
                e)))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               0 < x1 <= -((b d)/(-a + d))) || (b > 0 && 
               0 < x1 <= (-b d - b e)/(-a + d + e)))))))) || (d ==
     0 && ((a < 
        0 && ((α < 0 && b < 0 && 
           0 < x1 <= -((b e)/(-a + e))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b < 0 && 
           0 < x1 <= -((b e)/(-a + e))))) || (a == 
        0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 && 
           x1 > 0))) || (0 < a < 
        e && ((α < 
            0 && ((b < 0 && x1 >= -((b e)/(-a + e))) || (b >= 0 &&
                x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && x1 >= -((b e)/(-a + e))) || (b >= 0 &&
                x1 > 0))))) || (a == 
        e && ((α < 0 && b >= 0 && 
           x1 > 0) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b >= 0 && 
           x1 > 0))) || (a > 
        e && ((α < 0 && b > 0 && 
           0 < x1 <= -((b e)/(-a + e))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && b > 0 && 
           0 < x1 <= -((b e)/(-a + e))))))) || (d > 
    0 && ((a < 
        0 && ((α < 0 && 
           b < 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b < 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)))) || (a == 
        0 && α == 
        0 && ((b <= 
            0 && (0 < x1 < (-b d - b e)/(d + e) || 
             x1 > (-b d - b e)/(d + e))) || (b > 0 && 
           x1 > 0))) || (0 < a < 
        d && ((α < 0 && 
           b < 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b < 0 && (-b d - b e)/(-a + d + e) <= 
            x1 <= -((b d)/(-a + d))))) || (a == 
        d && ((α < 
            0 && ((b < 0 && 
               x1 >= (-b d - b e)/(-a + d + e)) || (b == 0 && 
               x1 > 0))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               x1 >= (-b d - b e)/(-a + d + e)) || (b == 0 && 
               x1 > 0))))) || (d < a < 
        d + e && ((α < 
            0 && ((b < 0 && 
               x1 >= (-b d - b e)/(-a + d + e)) || (b == 0 && 
               x1 > 0) || (b > 0 && 
               x1 >= -((b d)/(-a + d))))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b < 0 && 
               x1 >= (-b d - b e)/(-a + d + e)) || (b == 0 && 
               x1 > 0) || (b > 0 && 
               x1 >= -((b d)/(-a + d))))))) || (a == 
        d + e && ((α < 
            0 && ((b == 0 && x1 > 0) || (b > 0 && 
               x1 >= -((b d)/(-a + d))))) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 
            0 && ((b == 0 && x1 > 0) || (b > 0 && 
               x1 >= -((b d)/(-a + d))))))) || (a > 
        d + e && ((α < 0 && 
           b > 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e)) || (α == 
            0 && ((b <= 0 && (0 < x1 < -b || x1 > -b)) || (b > 0 &&
                x1 > 0))) || (α > 0 && 
           b > 0 && -((b d)/(-a + d)) <= 
            x1 <= (-b d - b e)/(-a + d + e))))))))*
$\endgroup$
1
  • $\begingroup$ thanks a lot for your help . $\endgroup$ Oct 3 at 7:08

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