Inequalities with a variable under a square root.

Question

Let's say I have an inequality. In order to solve it, I decide to multiple a square root on both sides - the number inside the root is a variable (x or y). Would I then need to switch the inequality signs. A square root is not a negative number per se, but it could be one and I might not know if it is so maybe I should switch the sign just to be safe?

Answer 1

Lets give an example to illustrate what's going on:

x² > 16

√x² > √16

x > ±4

We have "plus or minus" because 4 * 4 = 16 and -4 * -4 = 16

If you were to keep the sign as is, you would only get half of the answers. Similarly, if you flip the sign, you would have the other half of the answers.

This is where we go into a concept called absolute value. Absolute values are when we take a number line, and the number value is the distance away from 0. That is, 4 is 4 units away from 0, and -4 is also 4 units away from 0.

We use lines straight up and down to denote absolute values:

|x| > 4

Now, if you were to put a negative number in there, say, -5, then the absolute value would make it 5 (as that would be 5 units away from 0), and it would ring true in the final inequality, and if you place that into the original problem, you would get a statement that's also true:

-5 * -5 > 16

25 > 16

And that's how we can find the inequality answers to square roots.