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The final answer was 1

\( 8 < 4x + 4 < 2(x + 10)\\ 8 < 4x + 4 \quad and \quad 4x + 4 < 2(x + 10)\\ 4 < 4x \qquad and \qquad 4x + 4 < 2x + 20\\ 1 < x \qquad \;\;and \qquad \;\;\quad2x < 16\\ 1 < x \qquad \;\;and \qquad \;\;\quad x < 8\\~\\ 1

8 < 4x + 4 < 2(x + 10)

We have two inequalities to consider

8 < 4x+ 4                                                   4x +4  < 2(x + 10)

Subtract 4 from both sides                        divide through by 2

4 < 4x                                                         2x + 2 < x + 10

Divide both sides by 4                               Subtract 2, x from both sides

1 < x    (1)                                                           x  < 8   (2)

Combining (1) and (2) we have

1 < x < 8