To solve the inequality \(2^{x-1} > 5\), we can follow these steps:
1. Subtract 1 from both sides to isolate the exponent:
\[2^{x-1} - 1 > 4.\]
2. Rewrite the inequality using a common base:
\[2^{x-1} > 2^2.\]
3. Since the bases are the same, we can compare the exponents:
\[x - 1 > 2.\]
4. Add 1 to both sides to solve for \(x\):
\[x > 3.\]
Now, we have the solution \(x > 3\).
To graph the solution on the number line, draw an open circle at \(x = 3\) (since the inequality is not inclusive), and shade to the right to represent \(x > 3\).
Here's a textual representation:
```
<---|-------------------|-------------------|--->
0 3 4
```
The open circle at 3 indicates that \(x\) is greater than 3 but not equal to 3. The shaded region to the right represents all values of \(x\) greater than 3.
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