Video Transcript
State whether the following is true or false. If 𝑎 and 𝑏 belong to all reals and 𝑎 is greater than 𝑏, where 𝑎 and 𝑏 have the same signs, then one over 𝑎 is less than one over 𝑏.
We’ve been given quite a lot of information here. If 𝑎 and 𝑏 belong to all reals, they can be integers. They can be rational numbers or irrational numbers. They can be positive or negative. 𝑎 and 𝑏 cannot be imaginary numbers or ∞. We also say that 𝑎 and 𝑏 cannot be zero because they’ll either both be positive or both be negative.
The next thing we know is that 𝑎 is greater than 𝑏. We’re trying to say, is it true that if 𝑎 is greater than 𝑏, is one over 𝑎 less than one over 𝑏? One thing that might be helpful is for us to substitute in values for 𝑎 and 𝑏. Let’s say that 𝑎 is four and 𝑏 is two. We know that four is greater than two. Is one-fourth less than one-half? It is true that one-fourth is less than one-half?
We might wanna consider a second example where we’re working with negative values. We’ve been told that 𝑎 and 𝑏 both have to have the same signs. Remember 𝑎 needs to be greater than 𝑏. If we let 𝑎 be negative one, that’s greater than negative three. Both negative one and negative three have the same signs. One over 𝑎 would then be negative one over one, and one over 𝑏 would be negative one over three. And now, we have to think about is negative one less than negative one-third?
To think about this, let’s use a number line. If we have zero and negative one on the number line, where will negative one-third fall? Negative one-third will be closer to zero. And that means negative one-third is greater than negative one. In both cases, we’ve seen how if 𝑎 is greater than 𝑏 and 𝑎 and 𝑏 have the same signs, that one over 𝑎 will be less than one over 𝑏. This is a true statement.
