First, consider the inequality \(|2^{x+1} - 2| < 0.5\). This can be rewritten as two separate inequalities:

1. \(2^{x+1} - 2 < 0.5\)

2. \(2^{x+1} - 2 > -0.5\)

For the first inequality:

\(2^{x+1} < 2.5\)

Taking the logarithm base 2, we have:

\((x+1)\ln 2 < \ln 2.5\)

\(x < \frac{\ln 2.5}{\ln 2} - 1\)

For the second inequality:

\(2^{x+1} > 1.5\)

Taking the logarithm base 2, we have:

\((x+1)\ln 2 > \ln 1.5\)

\(x > \frac{\ln 1.5}{\ln 2} - 1\)

Calculating the critical values:

\(x < \frac{\ln 2.5}{\ln 2} - 1 \approx 0.322\)

\(x > \frac{\ln 1.5}{\ln 2} - 1 \approx -0.415\)

Thus, the solution is \(-0.415 < x < 0.322\).

Start with the inequality \(2(3^{1-2x}) < 5^x\).

Take the natural logarithm of both sides: \(\ln(2) + \ln(3^{1-2x}) < x \ln(5)\).

Apply the logarithm power rule: \(\ln(2) + (1-2x)\ln(3) < x \ln(5)\).

Rearrange the inequality: \(\ln(2) + \ln(3) - 2x \ln(3) < x \ln(5)\).

Combine like terms: \(\ln(6) < x(\ln(5) + 2\ln(3))\).

Solve for \(x\): \(x > \frac{\ln(6)}{\ln(5) + 2\ln(3)}\).

Simplify the denominator: \(x > \frac{\ln(6)}{\ln(45)}\).

To solve the inequality \(|3^x - 8| < 0.5\), we consider the two cases:

1. \(3^x - 8 < 0.5\)

2. \(3^x - 8 > -0.5\)

For the first case:

\(3^x < 8.5\)

Taking logarithms, \(x < \log_3(8.5)\)

For the second case:

\(3^x > 7.5\)

Taking logarithms, \(x > \log_3(7.5)\)

Calculating the logarithms to 3 significant figures:

\(\log_3(8.5) \approx 1.95\)

\(\log_3(7.5) \approx 1.83\)

Thus, the solution is \(1.83 < x < 1.95\).

To solve the inequality \(|2^x - 8| < 5\), we consider two cases:

1. \(2^x - 8 < 5\)

2. \(2^x - 8 > -5\)

For the first case, \(2^x - 8 < 5\):

\(2^x < 13\)

Taking logarithm base 2 on both sides, we get:

\(x < \log_2 13\)

For the second case, \(2^x - 8 > -5\):

\(2^x > 3\)

Taking logarithm base 2 on both sides, we get:

\(x > \log_2 3\)

Thus, combining both inequalities, we have:

\(\log_2 3 < x < \log_2 13\)

Calculating the logarithms, we find:

\(\log_2 3 \approx 1.58\) and \(\log_2 13 \approx 3.70\)

Therefore, the solution is \(1.58 < x < 3.70\).