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

Hi,

1 < e^5x − 1 < 4

First, take a Ln of inequality

Ln 1 < Ln  e^5x − 1< Ln 4

0 < 5x - 1 < Ln4 then add 1

0+1 < 5x-1 + 1 < 1 + Ln4

1 < 5x < 1 + Ln4 then divided by 5

1/5 < x < (1 + Ln4)/5

0.2 < x < 0.477 or interval notation is ( 0.2 , 0.477 )

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1 − 6 ln(x) < 13 minus 1

-1+ 1 -6 Lnx < 13 - 1

-6 Lnx < 12 divided by -6 and change < to >

Lnx > -2

as you know Ln e = 1 so Ln e ^-2= -2 back to the question and replace -2

Ln x > Ln e^-2 ------> x > e^-2= 0.135 ------> x > 0.135 and interval notation is (0.135, ∞ )

I hope it is useful,

Minoo

Answer 2

Take the ln of each term and solve