An Introductory Example

Solve `7 sqrt(|x|-1) = x+9`.

So `7 sqrt(|x|-1) = x+9 hArr x=-2 vv x=5 vv x=26`. What does this mean?

• `x=5`: both sides yield true
• `x=1`: both sides yields false
• `x=0`: the right hand side yields false, the left yields undefined

So we have `7 sqrt(|0|-1) = 0+9 hArr sf"F"`.

• When solving an inequation, we have `7 sqrt(|0|-1) le 0+9 hArr sf"F"`.
• By contraposition, do we also have `7 sqrt(|0|-1) > 0+9 hArr sf"T"`?
  We will not!

But it will not be trivial.

• Let us denote undefined with U.
• We have `sf"U" hArr sf"F"`.
• How can we avoid `sf"U" hArr not sf"U" hArr not sf"F" hArr sf"T"`?