To solve the equation $2^x + x = 5$, we'll use a combination of numerical and graphical methods, as there is no simple algebraic solution for this type of equation.

Step-by-Step Approach:

1. Graphing Method: Let's define two functions: $f(x) = 2^x + x$ and $g(x) = 5.$ We can graph both functions and find where they intersect, which will give us the value of $x$.

2. Trial and Error (Numerical Approximation): We can also estimate solutions by substituting values of $x$ into the equation $2^x + x$ to see which value satisfies the equation $2^x + x = 5$.

   • For $x = 1$: $2^1 + 1 = 2 + 1 = 3.$ This is less than 5.
   • For $x = 2$: $2^2 + 2 = 4 + 2 = 6.$ This is greater than 5.

   So, the solution lies between $x = 1$ and $x = 2$.

3. Refining the Estimate: We can try more precise values to narrow down the solution.

   • For $x = 1.5$: $2^{1.5} + 1.5 \approx 2.828 + 1.5 = 4.328.$ This is still less than 5.
   • For $x = 1.7$: $2^{1.7} + 1.7 \approx 3.249 + 1.7 = 4.949.$ This is very close to 5.
   • For $x = 1.72$: $2^{1.72} + 1.72 \approx 3.287 + 1.72 = 5.007.$ This is extremely close to 5.

Thus, the approximate solution is: $x \approx 1.72.$

Would you like more details or have any questions?

---

Here are 5 related questions:

1. How can we solve similar equations involving other exponential functions?
2. What are the different methods for solving transcendental equations?
3. How does graphing help in visualizing solutions to complex equations?
4. What is the significance of numerical methods in solving equations?
5. Can equations involving logarithms also be solved using graphical methods?

Tip: Always check for approximate solutions using different methods to confirm the result!