The equation 2^x = x + 5 can be solved by finding the x-coordinates where the graphs of y=2^x and y=x+5 intersect. How many real solutions…

Compare the graphs of $y = 2^x$ and $y = x + 5$. At $x = 3$, both equal 8, so one intersection is $(3, 8)$. For large $x$, the exponential grows faster than the line, so there are no more intersections to the right. For negative $x$, the line $y = x + 5$ decreases through zero at $x = -5$, while $2^x$ stays positive and small; at $x = -5$ the exponential is above the line ($\frac{1}{32} > 0$), but at $x = -4$ the line is above ($1 > \frac{1}{16}$). So they cross once between $-5$ and $-4$. That is 2 solutions, choice A.