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Solve[{-6 lx^2 (x - Subscript[lx, 1])^2 +
    12 lx (x - Subscript[lx, 1])^2 Subscript[lx, 1] -
    4 x^3 Subscript[lx, 2] +
    12 lx (x - Subscript[lx, 1])^2 Subscript[lx, 2] + 4
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    2 lx^2
\!\(\*SubsuperscriptBox[\(lx\), \(2\), \(2\)]\) -
    4 lx Subscript[lx, 1]
\!\(\*SubsuperscriptBox[\(lx\), \(2\), \(2\)]\) + 6
\!\(\*SubsuperscriptBox[\(lx\), \(1\), \(2\)]\)
\!\(\*SubsuperscriptBox[\(lx\), \(2\), \(2\)]\) - 4 lx
\!\(\*SubsuperscriptBox[\(lx\), \(2\), \(3\)]\) + 4 Subscript[lx, 1]
\!\(\*SubsuperscriptBox[\(lx\), \(2\), \(3\)]\) +
\!\(\*SubsuperscriptBox[\(lx\), \(2\), \(4\)]\) == 0,
  0 < Subscript[lx, 1] < x < Subscript[lx, 1] + Subscript[lx, 2] <
   lx}, x, Reals]
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    4*Subscript[lx, 1]^3*Subscript[lx, 2] + 2*lx^2*Subscript[lx, 2]^2 -
         4*lx*Subscript[lx, 1]*Subscript[lx, 2]^2 +
    6*Subscript[lx, 1]^2*Subscript[lx, 2]^2 - 4*lx*Subscript[lx, 2]^3 +
         4*Subscript[lx, 1]*Subscript[lx, 2]^3 + Subscript[lx, 2]^4 == 0,
  0 < Subscript[lx, 1] < x < Subscript[lx, 1] + Subscript[lx, 2] < lx}, x, Reals]

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\*UnderoverscriptBox[\(\[Sum]\), \(v = 1\), \(u\)]
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       {u, 1, i}]

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