P4 Chapter 1

etomac

how to do inequalities like this

eq.GIF1.5KB

Reply 1

Gaz031

etomac

how to do inequalities like this

Break it down into two seperate inequalities. Solve those, then the answer is the values of x which conform to both.
Be careful not to multiply/divide by something that can be negative or positive.

Reply 2

cms271828

I think I can improve slightly on the above answer.
Firstly its upto you if you decide to split the inequality or not.
There are generally 2 ways of dealing with values that can change sign.

1. Consider cases. so above let x>-2, then 2+x>0, so mulitplying out by 2+x
gives -2-x<2-x<=2+x, adding x gives -2<2<=2+2x, subtracting 2 gives 0<=2x,
thus x>=0
Since x was already >-2, then this range is ok.
Now let x<-2, then 2+x<0, so multiplying by 2+x gives 2+x<=2-x<-2-x
,adding x gives 2+2x<=2<-2, this is clearly false, so x<-2 must be invalid.
So the solution is x>=0.

2.The other more direct approach is to simply multiply by squares, since
squares are always >=0. In this example, we can multiply throughout by (2+x)^2
this then gives -(2+x)^2<(2-x)(2+x)<=(2+x)^2
=> -x^2-4x-4<-x^2+4<=x^2+4x+4
=> -4x-4<4<=2x^2+4x+4
=> -4<4+4x<=2x^2+8x+4
=> 0<4x+8<=2x^2+8x+8
=> 0<4(x+2)<=2(x+2)(x+2)
=> 0<4<=2(x+2) since x+2>0
=> 2<=x+2
=> x>=0.

Reply 3

etomac

thanks, both of u