Squaring inequalities

[yo4561]

MP's All the Quant Companion guide provides the following example:

If x^2<4, what are the possible values for x?

Then the answer says that if x is negative, then taking the square root is the equivalent of dividing by a negative, so you have to flip the inequality sign: x>-2.

To confirm, what is the rule for when one side of an inequality is negative? Is it just to flip the sign, as demonstrated in this example? I see that the chapter later discusses the rules for squaring inequalities but never touches upon when just one known side is negative and the other is positive.

Thank you

[esledge]

You know to beware of positive and negative solutions--that's the important thing. Rather than memorizing a bunch of rules about sign, here's what I do:

(1) Solve as if there's just an equal sign, so x^2 = 4 solves to x = 2 or -2.

(2) Sketch out a number line and draw some marks at x = 2 and x = -2. (Once you get the concept, you might not have to draw it.)

(3) Test a value (any value) on each side side of the marks, which will also include one between the marks. Plug into the original inequality, and put a note to yourself which range(s) work.

I'd probably test x = 0, x = -5 and x = +5. Only 0^2 < 4, and the other two values fail (because 25 is not less than 4), so that tells you the complete range of solutions is between -2 and 2. You only needed to test one value between the marks you drew, but if you tested any others, they would work too: -2 < x < 2.