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**terrorsquid** Show that there is exactly one real number $\displaystyle x$ for which$\displaystyle e^x + x = 5$. Find the integer part of this number. State any theorems you use in your solution.

My attempt:

$\displaystyle e^x+x-5 = 0$

$\displaystyle f(0) = 1-5 = -4$

$\displaystyle f(2) \approx 7+2-5 = 4$

By the intermediate value theorem, the continuous function crosses the x axis. The function is always increasing, therefore it crosses the x axis only once.

Is there an algebraic way for finding the integer part? I just subbed 1 into the original equation and got an answer < 5 then subbed in 2 and got an answer > 5 and deduced that x must be 1.something $\displaystyle \therefore$ the integer part is 1.