# Inequalities Are Not Broken Equations

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

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:

## 22 thoughts on “Inequalities Are Not Broken Equations”

1. Andrey says:

The interesting parts are the ideas, the useful parts are the inequalities.

2. What’s the secret to teaching quadratic and rational inequalities?

1. And, for quadratic inequalities in particular: (1) the graph is a parabola, does it open up or down (check the leading coefficient)—draw an appropriate curve w/o axes; (2) the quadratic formula can be used to find the zeroes—label appropriate points on the curve, then sketch in an \$x\$-axis; (3) read solutions to the inequality off of the graph.

1. While I love this, oddly, my students refuse to draw graphs for inequality problems!

1. I wrote a much longer comment, but it seems to have been eaten by the WordPress software. :\

In any event, we emphasize graphing in our precalculus curriculum here (a large public university). We teach students how to graph polynomials and rational functions (find the zeros and poles; characterize the behaviour near those special points; characterize the end behaviour; then connect the dots) before we mention inequalities, then use the graph for solving inequalities involving polynomial or rational functions.

For most “real world” applications, this is likely to be the best plan of attack. In particular, if you have a rational function which has not been factored, there is little hope of solving the problem by hand, but a computer can be used to graph the function and find the zeroes while still leaving the “difficult” task of correctly interpreting the graph to solve the inequality.

Alternatively, there are always sign tables (e.g. http://www.fmaths.com/signcharts/lesson.php).

2. I’ve been using sign tables for years without having a name for it. Though for some reason I draw out a number line instead of using a table. For more geometric flavor?

I try to teach as much graphing as I can, but due to time constraints, it’s not as much as I like. But nonetheless, students still seem to insist on solving the equation and then seemingly just guess at the solution to the inequality.

3. J.P. McCarthy says:

I endorse this comment 100%.

For rational inequalities multiply by squares (which are positive) to remove inequalities.

In general, if a function is increasing, we can apply the function to both sides of f(x) 10, we can take the log_2 of both sides.

4. First, I’ve been teaching the “add and subtract only” workaround for a long time as an admittedly clunky option. There are always a few students who appreciate having it as an out.

Second, I would not use the word “just” in the title here. It makes it sound like inequalities ARE broken equations, but also other things. And the title makes perfect sense without “just.” Most other modifiers that occur to me (merely, simply) have the same problem as “just.” And your claim not only loses the potential ambiguity without it but also gains “force.” Just my two cents. Which are less than three cents.

1. Persuaded! I’ll change it post haste.

1. I feel like I’ve impacted the long-term course of human thinking. 🙂

5. I can't think of a name says:

Small brain of sixth grader has completely exploded.

1. Ha! The “small brain” thing is my students’ favorite catch-phrase. You sound like one of them.

(…are you?)

6. Riccardo says:

“It says that x is for” -> four?

7. fred says:

Inequalities are so important in computing. Floating point numbers are really ranges, so you’re dealing with inequalities there. Most of the time, you can get away with ignoring it. But sometimes you can’t, and the math to avoid rounding getting out of hand gets real tough quickly. I wish more people were aware of the pitfalls
I think the same applies to measurements, but I don’t know enough to know for sure

8. When teaching, I found it helpful to separate the switching of the negative sign with the multiplication. Making the one step into two steps. eg: -5x 5x > -10 –> x > 2. I’ve found this tends to increase the flow of the lesson.

1. -5x 5x > -10 –> x > 2. WordPress messed with my equation.

1. again wordpress messed with the equation.

9. Saverio says:

My humble opinion: inequalities and equations should be treated as completely separate beast. They have a different purpose and they need different tools.

Inequalities should be treated with the same tool that are used to graph functions. Any inequality can be brought in the form somethingsomethingsomething > 0 (just by subtracting one side from the other one. That’s the only manipulation you need to do.
Then you need to study when somethingsomethingsomething is above zero, a problem that has different solutions in case you are dealing with linear functions of x, quadratic ones, etc.

10. I turn inequalities into equations because the equation defines the region boundary. Then, all you have to do is a simple test to figure out which side of the boundary the solution is on.

11. I never really understood inequalities in school. Now I do. At last! Thank you!