Be f: R ---> R, x ---> 2^x + x = exp(x.ln(2)) + x. For any real x we have f'(x) = (2^x).ln(2) + 1 > 0, so f strictly increases on R. Consequence: if the equation f(x) = 11 has a solution, then this solution is unique. As x = 3 is evident solution, then it is the only solution of the given equation.
You went through a lot of work for nothing. 11 = 2ˣ + x = 8 + 3 = 2³ + 3 = 2ˣ + x ⇒ x = 3 You were extremely lucky that 2³+3 = 11. If it was 2ˣ + x = 13, it might not have been solvable by hand.
Be f: R ---> R, x ---> 2^x + x = exp(x.ln(2)) + x. For any real x we have f'(x) = (2^x).ln(2) + 1 > 0, so f strictly increases on R.
Consequence: if the equation f(x) = 11 has a solution, then this solution is unique.
As x = 3 is evident solution, then it is the only solution of the given equation.
You went through a lot of work for nothing.
11 = 2ˣ + x = 8 + 3 = 2³ + 3 = 2ˣ + x ⇒ x = 3
You were extremely lucky that 2³+3 = 11. If it was 2ˣ + x = 13, it might not have been solvable by hand.
By inspection, x = 3. (Time taken = 2 seconds)