Hello again. I see the conceptual problem you're having, and with the help of my PChem textbook, I hope I can help clear it up. Prepare for a wall of text.
First of all, let's define what we mean by Vreal, Videal, Preal, Pideal, and so on. Videal is the volume that a given mole of ideal gas would occupy at a given pressure and temperature. Pideal is the pressure that a given mole of ideal gas would exert when it occupies a given volume at a given temperature. Vreal is the same as Videal, but for a real gas, and same goes for Preal. To compare Vreal to Videal, you have to hold everything else constant except for the identity of the gas (the factor that would make a gas not behave ideally).
Preal < Pideal and Vreal > Videal are both true statements only in strict circumstances. For the first one, you assume that the
only thing that separates real from ideal gases is intermolecular forces. For the second statement, you assume that the
only thing that separates real from ideal is that a real gas molecule occupies some volume, whereas an ideal gas molecule does not. Those assumptions have to be made for the statements to be true. Also, you might see that the two assumptions are mutually exclusive. If you have a gas that has significant intermolecular attractions, but insignificant volume taken up by the particles (i.e. the molecules are very small compared to the Videal so that Vreal = Videal), then you can say that Preal < Pideal. Similarly, if you have a gas that has significant volume taken up by the particles, but insignificant IMFs, then you can say that Vreal > Videal. If both factors are significant, then... we'll discuss that later. If both are insignificant, then the gas behaves ideally, of course.
The reason TBR and TPR state what appears to be opposite has to do with different terminology, and Van der Waal's equation. Forgetting about IMFs entirely and only considering the physical volume of the particles, it is true that the volume available will have to be less than if the particles behaved ideally and thus did not take up any volume (volume of the container). But here's the kicker. When TBR says Videal, they don't mean Videal in the sense I defined it earlier. They mean the volume that you would have to now plug into the ideal gas equation PV = nRT to get it to work out. The ideal gas equation becomes P(Vcontainer - Vparticles) = nRT. They define Vcontainer - Vparticles as Videal. TPR, however, defines Vcontainer as Videal, and (Vcontainer - Vparticles) as Vreal, so Vreal < Videal. Both of their approaches are not how I defined Videal and Vreal. To me, Vreal is the volume of the container of a real gas, because Vreal has to be measurable, and the only volume that is measurable is the volume of the container. To me, Videal is the volume of the container of a theoretical ideal gas under the same temperature and temperature as the real gas. The way Kaplan defines Vreal and Videal is the way I defined it, which is why tttgo's question makes no sense to you. By the way, (Vcontainer - Vparticles) is the same as V - nb in the Van der Waal's equation.
Now I will introduce a concept called compressibility factor. Z = pVm/RT, where Vm is molar volume (volume per mole). If a gas behaves ideally, then Z should be 1 at all times. Since R is constant, this also means that knowing two of the three variables (P, Vm, and T) is enough. The third variable is entirely determined by the other two. (A fancy, physical chemist's way of saying this is that the ideal gas law is an equation of state, but this is unimportant) For real gases, however, Z does not always equal 1. It turns out at low pressure and high temperature, Z approaches 1, and we can say that a real gas approaches ideal behavior as its pressure approaches 0 and its temperature approaches infinity. For most real gases, Z is below 1 if pressure is moderately high, but above 1 if pressure is extremely high. Going back to the definition of Z, this means that at moderately high pressure, pVm/RT is less than what would be expected by the ideal gas law. Since we've already prescribed a value for p, and R is a constant, that really means that Vm/T is less than what would be expected. At a given temperature, then this means that Vm is less than expected, or that
Vreal < Videal. Conversely, at extremely high pressure,
Vreal > Videal. What constitutes moderately high or extremely high is unique for each gas, and also dependent on temperature.
The physical explanation for this observation is that at moderately high pressure, the IMFs are significant, and thus bring the gas molecules closer together. This results in a reduction of volume (as I defined Vreal earlier) compared to what would be if there were no IMFs (Videal). At extremely high pressure, the gas molecules are so close together that the volume occupied by the particles begin to matter, so expansion of volume has to occur. Now imagine that a particular gas has no IMFs. Then Z would never go < 1, but it would still go > 1 as pressure increases. Since all gases, even the smallest possible (H2) has particles that take up definite volume, Z will eventually increase as pressure increases. For the TBR question, I think inert gas means a gas like helium that has practically no IMFs. In this case, Z is at minimum 1 (at the limit of P -> 0). As P increases, Z has to also increase, however slowly. Halving P means approaching 0, so Z has to decrease to 1, however little. Since Z is Vm/T, and we're presumably keeping T constant, then Vm cannot quite be twice the initial Vm. It has to be a little bit less.
I hope that helped. I really should use my spring break time with better things... Here are some wikipedia links to clear up confusions.
http://en.wikipedia.org/wiki/Van_der_Waals_equation
http://en.wikipedia.org/wiki/Compressibility_factor
