My algebra students and I recently wrapped up a unit on inequalities, which brought this old cartoon of mine back to mind:
I see why they do it. Working with equations is pretty straightforward: do the same thing to both sides, and they’ll remain equal.
Not so with inequalities. Sure, 5 > -4, but multiply both sides by -1, and suddenly you’ve got the claim that -5 > 4. If you believe that, then I’ve got a bridge I’d like to sell you for $4. (And then buy back for -$5. And then sell to you again.)
The point, as any algebra teacher can tell you: Inequalities reverse when you multiply (or divide) by a negative.
One way around this is the student sidestep, like so:
17 – x > 18 – 3x
17 – x = 18 – 3x
17 = 18 – 2x
-1 = -2x
0.5 = x
So far, so good. But now we’ve got to turn our result back into an inequality. We suspect the real answer is either x > 0.5 or x < 0.5, but we’ve got to check the possibilities back in the original inequality. It’s almost like having to solve the problem twice.
Instead, I prefer a different sidestep: the ads and subs only sidestep. Wherever possible, I try to avoid multiplication and division by negatives, like so:
17 – x > 18 – 3x
17 > 18 – 2x
-1 > -2x
(don’t divide by -2; instead, add 2x to both sides…)
2x – 1 > 0
(then, add 1 to both sides…)
2x > 1
(now, it’s safe to divide by 2)
x > 0.5
It can feel a bit like working with one hand tied behind your back. And it doesn’t cover all cases. Still, I strongly prefer it to the turn-it-into-an-equation sidestep, which feels to me like a betrayal of the whole spirit of inequalities!
Inequalities are a special resource to the mathematician, richer and more complex than those garden-variety creatures we call equations.
Take the inequality x > 4. It opens up conversations about the boundary line between solutions and non-solutions. In particular, 4 doesn’t work, but 4.00000000001 does. What’s the smallest solution? Does a “smallest” even exist?
By contrast, the equation x = 4 is a flavorless, gray nothingburger. It says that x is four, and there’s nothing more to add.
Perhaps that’s what Stephen Hawking had in mind when he gave this famous quip: