Question

how do you solve this equation

2 cos pi x = 4x + 5

Answer 1

Using a calculator set to radians, the old-fashioned trial-and-error method might work.
first I made a sketch of
y = 2cos πx and y = 4x+5 and noticed that there is one solution and x is negative

let f(x) = 2cosπx - 4x - 5
f(-1) = -.0015 , very close
f(-1.1) = +.39
f(-1.05) = .198
f(-1.025) = 0.984
f(-1.02) =.078
f(-1.01) = .0185
f(-1.0025) = .0085
f(-1.00125) = etc.

There is no simple arithmetic way to do this.
If you know Calculus, you might want to try Newton's Method