To solve the equation \(2^x + \frac{1}{2x} = 2.5\), we'll proceed with the following steps:
1. Rewrite the equation:
[tex]\[
2^x + \frac{1}{2x} = 2.5
\][/tex]
2. Isolate the exponential component:
Since \(2.5\) is equivalent to \(\frac{5}{2}\), we rewrite the equation as:
[tex]\[
2^x + \frac{1}{2x} = \frac{5}{2}
\][/tex]
3. Analyze the parts separately:
Let's first understand the behavior of \(2^x\) and \(\frac{1}{2x}\).
- \(2^x\) is an exponential function which increases as \(x\) increases.
- \(\frac{1}{2x}\) is a rational function which decreases as \(x\) increases (for \(x > 0\)).
4. Consider the possible values:
- When \(x = 1: \quad 2^1 + \frac{1}{2 \cdot 1} = 2 + 0.5 = 2.5\)
- Therefore, \(x = 1\) is a solution to the equation.
5. Verify the solution:
[tex]\[
2^1 + \frac{1}{2 \cdot 1} = 2 + 0.5 = 2.5
\][/tex]
Having successfully verified that \(x = 1\) satisfies the original equation, we have our solution:
[tex]\[
\boxed{1}
\][/tex]
Conclusion:
The solution to the equation [tex]\(2^x + \frac{1}{2x} = 2.5\)[/tex] is [tex]\(x = 1\)[/tex]. This detailed verification ensures that the solution is accurate and meets the requirements of the equation.
