Question

61-62: Solve each inequality for\(x\).

62. (a)\(1 < {e^{3x - 1}} < 2\) (b)\(1 - 2\ln x < 3\)

Short answer

(a) The solution of inequality is \(\frac{1}{3} < x < \frac{1}{3}\left( {1 + \ln 2} \right)\).

(b) The solution of inequality is \(x > {e^{ - 1}}\).

Step-by-step solution

Solve the inequality for \(x\)

\(\begin{aligned}1 < {e^{3x - 1}} < 2\\\ln 1 < 3x - 1 < \ln 2\\0 < 3x - 1 < \ln 2\\1 < 3x < 1 + \ln 2\\\frac{1}{3} < x < \frac{1}{3}\left( {1 + \ln 2} \right)\end{aligned}\)

Thus, the solution of inequality is \(\frac{1}{3} < x < \frac{1}{3}\left( {1 + \ln 2}

Solve the inequality for \(x\)

b)

Solve the inequality for \(x\) as shown below:

\(\begin{aligned}1 - 2\ln x < 3\\ - 2\ln x < 2\\\ln x > - 1\\x > {e^{ - 1}}\end{aligned}\)

Thus, the solution of inequality is \(x > {e^{ - 1}}\).