7 (x+4) - 2[x-3 (5+x) ]=⅔ (x-6)

7 (x+4) - 2/3 (x-6) < 2[x-3 (x+5) ]

Do it 1 step at a time.

7 (x+4) - 2/3 (x-6) < 2[x-3x-15]

7x+28 - 2/3 (x-6) < 2[x-3x-15]

I'll assume you mean (2/3) * x-6)

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7x+28 - (2x/3) - 4 < 2[-2x-15]

7x+24 - (2x/3) < - 4x-30

Add 4x

11x+24 - (2x/3) < - 30

Add 30

11x + 54 - (2x/3) < 0

Multiply thru by 3

33x + 162 - 2x < 0

31x < 162

x < 162/31