Solve each inequality for x. (Enter your answers using interval notation.)

Question

1 < e^5x − 1 < 4

and

1 − 6 ln(x) < 13

Answer 1

Solution:

1- 1 < e^5x-1< 4 Take ln of every side; ln 1 < ln e^5x-1 < ln 4, 0 < 5x-1 < ln 4

5x-1 > 0 add 1 to both sides 5x-1 +1 > 1, 5x > 1, x > 1/5, x >0.2

5x -1 < ln4 , 5x < 1n4 +1 , x < (ln 4+1) / 5 , x < (1.39 +1) / 5. x < 2.39 / 5, x < 0.48

The solution for x is 0.2 < x < 0.48 or 1/5 < x < (ln4 +1)/5

2- 1- 6ln(x) < 13, rearranging -6 ln(x) < 13-1, -6 ln(x) < 12, ln(x) < 12 /-6, ln(x) < -2 take exponential

of both sides, e^ln(x) < e^-2 , and x < e^-2 or 1/e²