# A trivial inequality, and how to express its solution in the most cryptic way imaginable

Solutions to olympiad problems are seldom written with clarity in mind — just look at forum posts in the Art of Problem Solving. The author makes jumps and skips a bunch of steps, expecting the reader to fill in the gaps.

Usually this is not much of a problem — the missing steps become obvious when you sit down and think about what’s going on with a pencil and some paper. But sometimes, this is not the case.

### The problem

One of the worst examples I’ve seen comes in the book Inequalities, A Mathematical Olympiad Approach. By all means, this is an excellent book. Anyways, here’s one of its easier problems — and you’re expected to solve it using the triangle inequality:

Prove that for all real numbers a and b, $||a|-|b|| \leq |a-b|$

### Attempt 1: Intuitive solution

It isn’t clear how the triangle inequality fits. If I weren’t required to use the triangle inequality, I might be tempted to do an intuitive, case-by-case argument.

Let’s visualize the absolute value of $a-b$ as the difference between the two numbers on a number line. Now we compare this distance $|a-b|$ with the distance after you take the absolute value of both of them, $||a|-|b||$. If one of the numbers is positive and the other negative, we clearly have a smaller distance if we ‘reflect’ the negative one over. Of course, if they’re both positive, or they’re both negative, then nothing happens and the distances remain equal.

There, a simple, fairly clear argument. Now let’s see what the book says.

### The book’s solution

Flip to the end of the book, and find

Consider $|a|=|a-b+b|$ and $|b|=|b-a+a|$, and apply the triangle inequality.

Huh. Perhaps if you are better versed than I am in the art of solving inequalities, you’ll understand what this solution is saying. But I, of course, had no idea.

Maybe try the substitution they suggest. I only see one place to possibly substitute $|a|$ for anything — and substituting gives $||a-b+b|-|b-a+a||$. Now what? I don’t think I did it right — this doesn’t make any sense.

To be fair, I cheated a little bit in the first attempt: I didn’t use the triangle inequality. Fair enough — let’s solve it with the triangle inequality then and come back to see if the solution makes any sense now.

### Attempt 2: Triangle inequality solution

A standard corollary to the triangle inequality of two variables is the following: $|a|-|b| \leq |a-b|$

Combine this with the two variables switched around: $|b|-|a| \leq |b-a| = |a-b|$

Combine the two inequalities and we get the desired $||a|-|b|| \leq |a-b|$

Now let’s look at the solution again. Does it make sense? No, at no point here  did we do any $|a-b+b|$ substitution. Clearly the authors were thinking of a different solution that happened to also use the triangle inequality. Whatever it was, I had no idea what the solution meant.

### The book’s solution, decrypted

Out of ideas and hardly apt to let the issue rest, I consulted help online at a math forum. And look — it turns out that my solution was without a doubt the same solution as the book’s intended solution!

What the author meant was this: considering that $|a| = |a-b+b|$, we have $|a| \leq |a-b|+|b|$ from the triangle inequality. Then, moving the $|b|$ over we get $|a|-|b| \leq |a-b|$.

After that, the steps I took above are left to the reader.

Perhaps I’m a bit thick-headed, but your solution can’t possibly be very clear if a reader has the exact same solution yet can’t even recognize your solution as the same solution. Come to think of it, if I couldn’t even recognize the solution, what chance is there of anybody being able to follow the solution — especially if they’re new to inequalities?

Almost every one of the one-sentence phrasings of this solution I could think of would be clearer and less puzzling than the solution the book gives me.

## 2 thoughts on “A trivial inequality, and how to express its solution in the most cryptic way imaginable”

1. Larae says:

2. journeyinmath says: