# Squares and inequalities

1. Feb 3, 2010

### TyErd

Alright lets just say (x-2)^2>12, find x

can someone tell me what happens to the inequality sign when you take the square of the left hand side to the right hands side? does it swap?

2. Feb 4, 2010

### Staff: Mentor

Let's think about a simpler problem instead. The way you are approaching this, you will almost certainly get the wrong answer or miss half of them

Let's try a2 > 4. What is the solution set for this inequality?

3. Feb 4, 2010

-2<a<2

4. Feb 4, 2010

### TyErd

so with these types of questions you need to draw the graph first?

5. Feb 4, 2010

### TyErd

no im wrong its a<-2 and a>2

6. Feb 4, 2010

### Staff: Mentor

You're almost spot on: It's a < - 2 OR a > 2.

The idea is that if a2 > 4, then a is larger than 2 or a is more negative than -2. In symbols this is a > 2 or a < - 2.

Now for the problem you asked...
(x - 2)2 > 12
then x - 2 > ? or x - 2 < ??
If you get these right, all that remains is to add + 2 to both sides of each inequality.

7. Feb 4, 2010

### TyErd

x-2>sqrt12 OR x-2<-sqrt12

thus x>sqrt12+2 OR x<-sqrt12+2??

8. Feb 4, 2010

### Staff: Mentor

Right. Another way to write the solution is x > 2 + 2sqrt(3) or x < 2 - 2sqrt(3). Both ways are correct, though.

Now if my problem had been a2 < 9, then a has to be smaller than 3 (but not too small -- i.e., not too negative) AND a has to be larger than -3 (but not too large).

So a < 3 and a > - 3. This means that a is any number between -3 and + 3. This is usually written as a continued inequality, with the smallest number on the left and the largest on the right: -3 < a < 3.

You could write this as 3 > a > -3, and it means the same thing, but this is not used as much.

9. Feb 4, 2010

### TyErd

okay thankyou heaps. This happened to be part of my final year exam practice paper. I would have been screwed if I hadnt known this. Thanks!