CV, EV for additive utility; confirm or deny
Questions with numbers are usually not as good as questions without numbers. If you had written down the formula for CV and EV you would probably have noticed that your premise is false.



CV and EV are not supposed to have opposing signs. You can see this from their definitions where
$$
CV = e(p_1,u_1) - e(p_1,u_0), \hskip 20pt EV = e(p_0,u_1) - e(p_0,u_0).
$$
Either $u_1 &gt; u_0$ and then a larger income is needed to reach $u_1$ given any price or the opposite is true.



In your final comment you also note that good 2 is inferior. As can be seen from your demand function

$$
x^*(p, w) = \left(\frac{p_2 w}{p_1(4p_1 + p_2)} , \frac{4p_1 w}{p_2(4p_1 + p_2)}\right)
$$

this is not true. If income were to increase $x_2^*(p, w)$ would increase as well.