If log 4 ( 3*4x+1 -8) ≤ 2x+1, then ( 3*4x+1 -8) ≤ 4(2x+1)
or, 4(2x+1) - 3*4x+1 ≥ -8
or, 4x+1(4x -3) ≥ -8
Now, let 4x = y. Then the above inequality changes to 4y(y-3) ≥ -8
or, 4y2 -12y+8 ≥ 0 . Now, on diving by 4, we get y2 -3y+2 ≥ 0 or, (y-2)(y-1) ≥ 0
Now, there are 2 possibilities as under:
1. y -1 ≥ 0 and y -2 ≥ 0 so that y ≥ 2.
2. y -1 ≤ 0 and y -2 ≤ 0 so that y ≤ 1 .
If y ≥ 2, then 4x ≥ 2 or, 22x ≥ 21 so that 2x ≥ 1 or, x ≥ 1/2.
If y ≤ 1, then 4x ≤ 1 or, 4x ≤ 40 so that x ≤ 0.
Hence, the answer is x ≤ 0 or, x ≥ 1/2.
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