Hi, Beautiful Mind. I think you have done great!

a) . The original inequality says that could be any number greater than and less than ; and that includes, for example, the number , which the doesn't cover. On the other hand says that could be any number between and . This includes the numbers and which are not true for . Thus is not necessarily true. It couldpossiblybe true though, because both the inequalities can take the values and , for example. But it is notnecessarilytrue.

b) . It says that when , which is greater than and less than , is substracted by , its value falls between and . The original inequality was . We want to find the region for which holds the truth for , when is greater than and less than . The result which we then get is the only the one which the inequality holds true for . Thus if what we find doesn't turn out to be , then the inequality is not necessarily true. If, on the other hand, we find to be the region for which holds for , when is greater than and less than , then the inequality is necessarily true. To proceed, since we need to find the inequality for , it's necessary that we have to substract from all the parts of the inequality; it's by rule that what is done to one side of an inequality should as well be done to the other. So we have , which is gives . Hence is necessarily true.

c) . This says when a value of greater than and less than is divided by the answer will fall between and . Ignore this for a minute and remember our original inequality. It was , basically saying that is greater than and less than . Now, you want to find the region for which the inequality is true is true for when is divided by . Since, by rule, whatever you do to one side of an inequality or an equation should as well be done to the other, you divide all the sides of the inequality by . That is which give you . Thus is necessarily true.

d) . This says when you take the reciprocal of {when you divide by }, where , as the original inequality holds, is greater than and less than , the value of lies between and . To check this, let us start with the original inequality. . Since we need to find the inequality which holds true for the reciprocal of {when is divided by }, and since by rule anything done to one side of an inequality should as well be done to the other, we have to take the reciprocal of the whole inequality. This is done by dividing 1 by the whole inequality. So we have . Now, if you take the reciprocal of an inequality, the sign of the inequality reverses. Thus , which is another way of writing , since both mean that lies between and . Thus is necessarily true.

e) . This is actually similar to the above. You take the same steps; divide the by the original inequality, and then if the result that comes out is , the inequality ( ) is necessarily true; if that doesn't come up, then it isn't. So, let us proceed:

.

Therefore is necessarily true.