4) There are at least three methods.
Method 1: When two variables are inversely proportional, their *product* is constant.
So temp1 * volume1 = temp2 * volume2; 30(90) = temp2*(18).
The new temperature is temp2 = 30(90)/18 = 150°C.
Method 2: When two variables are inversely proportional, the ratio of two values of one variable is the reverse of the ratio of the corresponding values of the other variable.
The ratio of the old to new volumes is 90:18 or 5:1, so the ratio of the old to new temperature is 1:5.
So the new temperature is 30(5) = 150°C.
Method 3: Let y = temperature and x = volume.
Since y varies inversely with x, y = k/x for some constant k.
When x = 90, y = 30, so 30 = k/90 and then k = 2700. Thus y = 2700/x.
When x = 18, y = 2700/18 = 150°C.
(Note that I solved this problem as it was stated in your question. In the real world, in order for the temperature of an ideal gas to vary inversely with the volume while the mass remains constant, the pressure would simultaneously have to vary inversely with the square of the volume.)
5) I will assume that you mean y = 7/(2x + 48) and that excluded values of x are needed.
The excluded values are the values of x that make the denominator 2x + 48 equal to zero.
2x + 48 = 0; 2x = -48; x = -24.
So the excluded value is -24.
Lord bless you today!