To solve the equation \( 2^x + x = 5 \), let's go through the steps.

### Step 1: Isolate the Exponential Term
The equation is already in a form where the exponential term \( 2^x \) and the linear term \( x \) are separated, so we can proceed by testing possible values of \( x \).

### Step 2: Test Simple Values of \( x \)
Let's try some simple values for \( x \):

- For \( x = 0 \):
  \[
  2^0 + 0 = 1 + 0 = 1 \quad (\text{Not } 5)
  \]

- For \( x = 1 \):
  \[
  2^1 + 1 = 2 + 1 = 3 \quad (\text{Not } 5)
  \]

- For \( x = 2 \):
  \[
  2^2 + 2 = 4 + 2 = 6 \quad (\text{Greater than } 5)
  \]

Since \( 2^x + x \) is increasing and we found that \( x = 1 \) gives 3 (less than 5) and \( x = 2 \) gives 6 (greater than 5), there must be a solution between \( x = 1 \) and \( x = 2 \).

### Step 3: Use Numerical Methods
To find the exact solution, we can use a numerical method like the **bisection method** or **Newton's method**. However, for simplicity, let's approximate it by trying values between 1 and 2.

- For \( x = 1.5 \):
  \[
  2^{1.5} + 1.5 \approx 2.828 + 1.5 = 4.328 \quad (\text{Less than } 5)
  \]

- For \( x = 1.7 \):
  \[
  2^{1.7} + 1.7 \approx 3.249 + 1.7 = 4.949 \quad (\text{Close to } 5)
  \]

- For \( x = 1.75 \):
  \[
  2^{1.75} + 1.75 \approx 3.363 + 1.75 = 5.113 \quad (\text{Slightly greater than } 5)
  \]

We see that the solution is very close to \( x = 1.7 \), and a more precise value using a calculator or a numerical method gives:

\[
x \approx 1.71
\]

### Final Answer:
The solution to the equation \( 2^x + x = 5 \) is approximately \( x \approx 1.71 \).

Would you like more details on how to solve it numerically or have any questions?

### Related Questions:
1. How would you use the bisection method to find the root of this equation?
2. Can you solve \( 3^x + x = 10 \) using a similar approach?
3. What happens if you change the base in the equation \( 2^x + x = 5 \) to another number, say 3?
4. How can Newton's method be applied to find the root of this equation?
5. Can you graph the function \( f(x) = 2^x + x - 5 \) and explain the root visually?

### Tip:
When dealing with equations that mix linear and exponential terms, numerical methods are often the best approach to finding precise solutions.

Learn how to solve the equation 2^x + x = 5 with a detailed step-by-step solution. This problem covers exponential functions, numerical methods like the bisection method and Newton's method, and equation solving techniques. Suitable for advanced high school students and anyone interested in mathematical problem-solving.

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