John H said:Homework Statement
Prove that if b is a positive number, such that a < b, then
a/b<(a+1)/(b+1)
2. The attempt at a solution
I have tried a few things, attempting to prove it using the real line, and a bunch of other methods but have had no success. I Would greatly appreciate it if u can at least get me started.
The purpose of proving inequality is to demonstrate that a certain statement or equation is always true, regardless of the specific values of the variables involved. This allows us to make generalizations and draw conclusions about a larger set of numbers or situations.
The notation a/b < (a+1)/(b+1) means that the value of a/b is always less than the value of (a+1)/(b+1). This is represented by the < symbol, which means "less than". The a and b represent variables, which can take on any numerical value. The inequality is also restricted to cases where b > a, meaning that the value of b must always be greater than the value of a.
The approach used is called mathematical induction, which is a method of proof that involves showing that a statement is true for a specific case (in this case, when a = 1 and b = 2), and then proving that if it is true for a given case, it must also be true for the next case (in this case, a = k and b = k+1).
Specifying that b > a is necessary because without this restriction, the inequality may not hold true. For example, if b = 1 and a = 2, then a/b is greater than (a+1)/(b+1). Therefore, in order to prove the inequality, we must specify that b > a.
Yes, this inequality can be applied to all values of a and b as long as b > a. This includes all positive, negative, and fractional values of a and b. However, it is important to note that the proof only holds true for b > a, and if this condition is not met, the inequality may not hold true.