To solve the equation \(x + 4 = x - 2^x\), let's start by simplifying it:

1. Subtract \(x\) from both sides:
   \[
   4 = -2^x
   \]

2. Multiply both sides by \(-1\):
   \[
   -4 = 2^x
   \]

Now, observe that \(2^x\) is always positive for any real value of \(x\). Since \(-4\) is negative, there is no real value of \(x\) that will satisfy the equation \(2^x = -4\).

**Conclusion:**  
There is no real solution to the equation \(x + 4 = x - 2^x\).

Would you like details on why \(2^x\) cannot be negative, or do you have any other questions?

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Here are five related questions to explore further:

1. What is the domain of the function \(f(x) = 2^x\)?
2. Can you solve \(2^x = 0\) for any real value of \(x\)?
3. How do the graphs of \(y = 2^x\) and \(y = -2^x\) differ?
4. What happens to \(2^x\) as \(x\) approaches infinity and negative infinity?
5. Can you solve \(x + 4 = x - 3^x\) similarly?

**Tip:** When dealing with exponential functions like \(2^x\), remember that they are always positive for any real number \(x\).

Learn how to solve the equation x + 4 = x - 2^x step-by-step with a detailed explanation. This problem involves exponential functions and requires understanding the behavior of exponential expressions in equations. Suitable for high school students.

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