To solve the equation \( 2^{x} + x = 5 \), we have to find the value of \( x \) that satisfies this equation. This equation combines both an exponential term and a linear term, making it challenging to solve algebraically. Therefore, we'll solve it using numerical methods or by estimating the solution.

### Steps:
1. **Graphical Method**: One way to solve this is by plotting the functions \( y = 2^x + x \) and \( y = 5 \) on the same graph and finding their intersection.

2. **Numerical Approximation**: We can also estimate the solution using trial and error or more sophisticated numerical methods like Newton's method.

### Estimating Solution:
Let's start by checking some values of \( x \):

- For \( x = 0 \): 
  \[
  2^0 + 0 = 1 + 0 = 1
  \]
  which is less than 5.

- For \( x = 1 \):
  \[
  2^1 + 1 = 2 + 1 = 3
  \]
  which is still less than 5.

- For \( x = 2 \):
  \[
  2^2 + 2 = 4 + 2 = 6
  \]
  which is greater than 5.

Since the function \( 2^x + x \) is continuous and increasing, there must be a value of \( x \) between 1 and 2 where the equation holds.

### Refined Estimation:
Let's try \( x = 1.5 \):
\[
2^{1.5} + 1.5 \approx 2.828 + 1.5 = 4.328
\]
which is still less than 5.

Now, let's try \( x = 1.7 \):
\[
2^{1.7} + 1.7 \approx 3.249 + 1.7 = 4.949
\]
which is very close to 5.

Finally, let's try \( x = 1.8 \):
\[
2^{1.8} + 1.8 \approx 3.482 + 1.8 = 5.282
\]
which is slightly greater than 5.

Thus, the solution is approximately \( x \approx 1.7 \).

Would you like more details on the numerical method, or do you have any other questions?

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Here are five related questions for further exploration:

1. How can you use Newton's method to refine the estimate of \( x \)?
2. What is the general approach to solving transcendental equations like this one?
3. How would the solution change if the equation were \( 2^x + x = 6 \)?
4. What if the equation were \( 3^x + x = 5 \) instead?
5. Can this equation be solved using logarithms?

**Tip:** For equations involving both exponential and linear terms, numerical methods or graphing are often the best approach to finding solutions.

Explore how to solve the equation 2^x + x = 5 using graphical methods and numerical approximation. This problem involves exponential functions and requires methods like plotting graphs and trial estimation. Suitable for students in Grades 10-12.

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