Solve each inequality for x. (Enter your answers using interval notation.)(a)1 < e9x − 1 < 6

To solve the inequality 1 < e^(9x − 1) < 6, we can break it down into two separate inequalities and solve them individually:

1. 1 < e^(9x − 1)
2. e^(9x − 1) < 6

Let's solve each inequality:

1. For 1 < e^(9x − 1), we can start by taking the natural logarithm (ln) of both sides to get rid of the e on the right side:

   ln(1) < ln(e^(9x − 1))

   Since ln(1) = 0, we have:

   0 < 9x - 1

   Solving for x gives us:

   x > 1/9

2. For e^(9x − 1) < 6, we again take the natural logarithm of both sides:

   ln(e^(9x − 1)) < ln(6)

   This simplifies to:

   9x - 1 < ln(6)

   Solving for x gives us:

   x < (ln(6) + 1) / 9

Therefore, the solution to the inequality 1 < e^(9x − 1) < 6 in interval notation is (1/9, (ln(6) + 1) / 9).