Inequalities Involving > and ≥

Once again let's begin along a simple number example.

                                                    p ≥ 4

It says that whatever p is it has to be at least a distance of 4 from the origin and thus p has to be in one of the following two ranges,

                                 p ≤ -4                         or                                        p ≥ 4

Before giving the general solution we have to address a common mistake which students make with these types of problems.  Several students try to join these into a single double inequality as follows,

                                                          -4 ≥ p ≥ 4

Whereas this might appear to make sense we can't stress sufficient that THIS IS NOT CORRECT!! Remind what a double inequality says.  In a double inequality we need that both of the inequalities be satisfied simultaneously. Then the double inequality above would mean that p is a number which is simultaneously smaller than -4 and larger than 4. It just doesn't make sense. There is no number which satisfies this.

These solutions have to be written as two inequalities. Here is the general formula for these.

If         p          ≥ b, b = 0         then     p ≤ -b   or  p ≥ b

If         p          > b, b = 0         then     p < -b   or  p > b