# How do we find the least upper bound and greatest lower bound?

## Homework Statement

1. (a) Solve the following inequalities and express the solutions first in interval notation, then
express those intervals in set builder notation.

(i) $$x3 + x2 > 2x$$ (ii) $$\left|(2-x)\right| \leq 4$$ .

(b) For each of the solution sets in part (a), state the least upper bound and greatest lower bound,
if these exist, or say they do not exist.

N/A

## The Attempt at a Solution

I have found that:
For (i) x is between -2 and 0 or x greater than 1.
For (ii) x is between -2 and 6 (including -2 and 6)

Are the bounds just the extreme values of the domain that the function can take? I just want to make sure.

Thanks,
Charismaztex

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Are the bounds just the extreme values of the domain that the function can take? I just want to make sure.
Not exactly. The least upper bound and greatest lower bound need not be part of the set at all. The greatest lower bound is just the greatest value such that every element of the set is greater than it. For example, in your first problem, the greatest lower bound of the solution set is -2, but -2 is not in the domain.

Thanks :)