I'm looking for real solutions to the following equation: $$2^x = \frac{3}{4}x + \frac{5}{4}$$ By plotting the functions I've found that the solutions should be $\pm 1$, however I this approach is not viable for solving similar problems in the future.
I'm curious, how would one approach an equation such as this algebraically, since I have no prior experience with equations of similar structure.
$f(x)=2^x$ is a convex function, which says that the line $y=\frac{3}{4}x+\frac{5}{4}$ and a graph of $f$ have two common points maximum.
But by your work $1$ and $-1$ are roots of the equation, which says that our equation has no another roots.
This way does not help in the general, if you want to get roots in elementary functions.