What happens to the inequality sign when taking the square of an equation?

[TyErd]

Alright let's 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?

 

[Mark44]

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?

 

[TyErd]

-2<a<2

 

[TyErd]

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

 

[TyErd]

no I am wrong its a<-2 and a>2

 

[Mark44]

no I am wrong its a<-2 and a>2

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.

 

[TyErd]

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

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

 

[Mark44]

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.

 

[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!