Can a system of two linear inequalities only have one solution?

Can Two Simple Inequalities Really Have Just One Answer?

So, you’re wondering if a pair of linear inequalities can box you into a single, solitary solution? It’s a great question, and the answer, while a bit technical, is actually pretty cool when you break it down.

First off, let’s quickly recap what we’re even talking about. Linear inequalities aren’t like those neat, tidy equations you remember from algebra class. Instead of pinpointing exact values, they paint a picture of ranges of values. Think “greater than,” “less than,” or even “greater than or equal to.” Now, when you throw two or more of these inequalities into the mix – bam! – you’ve got a system of linear inequalities. The solution? It’s basically the sweet spot, the collection of all the points that make every inequality happy at the same time.

Graphically, this sweet spot becomes a shaded region on a graph. Each inequality carves out its own territory, and where those territories overlap, that’s where you find your solutions.

Now, here’s where it gets interesting. The shape of that solution region is always… well, it’s a bit of a mouthful: “convex.” What does that even mean? Imagine drawing a line between any two points inside that shaded area. If the entire line stays inside the shaded area, then you’ve got a convex region. Think of it like a slightly inflated balloon – no dents or pinches.

And that’s why it’s so tough to get just one solution. Because of this “convexity” thing, you usually end up with:

• Nada. Zilch. No solution: The shaded areas just don’t meet. They’re like two ships passing in the night.
• A gazillion solutions: The shaded areas overlap, creating this big, unbounded convex region with endless possibilities.

But wait! Is there a loophole? Could we possibly squeeze out just one single solution?

Okay, so picture this: Imagine the lines that define your inequalities perfectly intersect at a single point. And that point just so happens to satisfy both inequalities. Technically, in that super-specific, almost-too-good-to-be-true scenario, you could argue that you have only one solution.

But here’s the kicker: it’s incredibly fragile. Breathe on it wrong, change one tiny thing – like switching “greater than” to “greater than or equal to” – and suddenly, you’re swimming in infinite solutions again. The slightest change blows the whole thing up!

So, bottom line? While it’s theoretically possible to trap your inequalities into a single solution, it’s about as likely as finding a unicorn riding a skateboard. In the real world, you’re almost always going to end up with either no solutions at all or a whole lot more than you bargained for. The convex nature of inequalities just makes sure of it.