describe the domain of a function $f(x, y)$ Describe the domain of the function $f(x,y) = \ln(7 - x - y)$. I have the answer narrowed down but I am not sure if it would be $\{(x,y) \mid y ≤ -x + 7\}$ or $\{(x,y) \mid y < -x + 7\}$ please help me.

Nice job: on the inequality part of things; the only confusion you seem to have is with respect to whether to include the case $y = x-7$. But note that $$f(x, y) = f(x, x - 7) = \ln(7 - x - y) = \ln \left[7 - x -(7 - x)\right] = \ln 0$$ but $\;\ln (0)\;$ is not defined, hence  $y = x - 7$ cannot be included domain!
So we want the domain to be one with the strict inequality: 
$$\{(x,y) | y < -x + 7\}$$
