Are rational solutions and real solutions the same thing?

No, because real solutions could involve irrational numbers as well as rational numbers.

As a society, we no longer encourage people to get a good education, work hard, and better themselves. Instead, we cater to the lowest-common denominator; allowing parents to sue schools when their kids fail, have school boards debating the 'validity' of ID in classrooms, and lower the standards at every turn. Little Johnny's self esteem is not my problem. If the kid is a dumbas$, he's a dumbas$; and the rest of the class shouldn't have to lower and limit their ability in order to help him feel better about it. Maybe if we quit accepting poor performance (on all levels) we might be able to drag ourselves back into a more advanced position. As for the home-schooling aspect: the only way that this should be allowed, is if the parent doing the 'teaching' can pass the same same certifications that their public school counterparts have to. If little Johnny's dad/mom can't provide a similar or more advanced education than what the public school's in their community do, get the kid back into a classroom. Home schooled kids should be held to the same educational standards, as well. School districts and states have fairly specific curriculums that the schools must adhere to, as well as standardized tests to gauge where the problems are. Many states treat home-schooling as different than public (or private schools), and that needs to stop.

Not at all. Rational solutions consist of rational numbers, which are final, as you may or may not know. The "final" means that it can be presented in the form of a fraction. (1/4, 5, 27478554, 2,22222..., 0.9898...). Real solutions, however, consist of both rational and irrational numbers. Irrational numbers are those who cannot be presented in the form of a fraction (as a decimal number, they don't seem to have a pattern (0.3476543..., 54,848792348...))

The set of rational numbers are a subset of the set of real numbers.

Rational numbers can be expressed as a fraction. 0.1, 2/3 and recurring decimals like 0.3333333 are examples. All rational numbers are real.

But not all real numbers are rational i.e. some real numbers cannot be expressed as a fraction, e.g. pi and e. These also cannot be expressed as a recurring decimal.