Ask Uncle Colin: An Unclear Inequality

Dear Uncle Colin,

I solved $(x+1)(x-2)(x+3)>0$ by saying there were three possibilities, $x+1>0$, $x-2>0$ or $x+3>0$. The middle one gives $x>2$ and that’s the strictest, so that was my answer - but apparently it’s wrong. Why is that?

- Logical Expressions Seem Silly

Hi, LESS, and thank you for your message!

I have one piece of advice for you, one:

Sketch. Sketch, sketch, sketch.

I though it was such an important piece of advice that it was worth repeating four times.

The most critical thing you can do when working with an inequality is to sketch it. See where it gets involved with the axes. See what it does when $x$ is big (positively and negatively). See about turning points and asymptotes, if appropriate. Get the shape right.

Then, only then, should you try to answer the question.

With this one

Sketching $y=(x+1)(x-2)(x+3)$ gives a cubic that crosses the $x$-axis at $(-3,0)$, $(-1,0)$, and $(2,0)$. It also crosses the $y$-axis at $(0,-6)$, which is enough for us to sketch the graph: it comes up from below, crosses the axis upwards at $(-3,0)$, then turns around to cross downwards at $(-1,0)$, then turns again to come upwards through $(2,0)$.

And we want to know where the $y$-coordinate is positive.

Now we have the sketch, that’s obvious: it’s not just the $x>2$ that you found (slightly dubiously), but also $-3<x<-1$.

Hope that helps,

- Uncle Colin