# Difference of inequations

I'm currently studying for a class where the teacher's notes were given, but there are many errors here and there so I need to make sure that everything on it is correct. I'm given the following system of inequations:

$$\left\{\begin{matrix} 0 \leq t'_1 + t_2 \leq 10 \;\;\;\;\; (a) \\ 5\leq t_2 \leq 15 \;\;\;\;\;\;\;\;\;\;\;\; (b) \\ 12\leq t'_3+t_2\leq 22 \;\;\; (c) \end{matrix}\right.$$

Now I need to eliminate t2 form (a) and (c) so on the notes here is the resulting system:

$$\left\{\begin{matrix}-5 \leq t'_1 \leq 5 \;\;\;(a+(-b)) \\2\leq t'_3 \leq 17\;\;\; (c+(-b)) \\5\leq t_2\leq15 \;\;\;\;\;\;\;\;\;\;\;\;\;\;(b) \end{matrix}\right.$$

Now the question is: can someone guide me through the process of eliminating t2 ? Are the resulting inequations actually correct ?

To eliminate $t_2$, first we multiply the second inequation by $-1$ and then we add it to the first one:
$$5 \leq t_2 \leq 15 \implies -5 \geq -t_2 \geq -15$$ We need to flip the inequation signs since we're multiplying by a negative number.
Now, we have a problem here as we can't add 2 inequations if the inequation signs are opposite; we can only subtract them. Since subtracting them is equivalent to just adding them before multiplying by $-1$, we see that either way this won't work in eliminating $t_2$. The notes are definitely wrong; they forgot to flip the inequation signs. (Don't even ask me about the havoc they wreaked on the last inequation; for starters, $(b)$ isn't even the same inequation in the top than in the bottom!)