Inequalities (1 Viewer)

[NickP101]

Hey all, just something i dont understand.

Take 1/(x-2) <= 3 for example.

Now as we dont know if the denominator on the LHS is <0 we need to multiply by (x-2)^2 on all sides. Thats fine i understand this.

I was stuffing around the other night and i made a stupid mistake, i squared the entire LHS and the RHS. Leaving me with the eqn 1/(x-2)^2 <= 9.

Now i simply multiples through by (x-2)^2 and was on my way, ofcoarse i got the incorrect answer.

What i dont understand is why the latter produces an incorrect answer? Im squaring both sides so shouldnt the eqn remain =? I know there is something wrong with my thinking. Can anyone see where ive made a mistake in my thinking?

Sorry for the easy question, its just bugging me.

Any help would be greatly appreciated.

Nick.

 

[SoulSearcher]

Ok you have your original inequality 1/(x-2) < 3
multiply by (x-2)²
(x-2) < 3(x-2)²,
(3x-7)(x-2) > 0 and then you figure out the answer from there,
but if you square the original inequality, it becomes 1/(x-2)² < 9, thus altering the original inequality.
multiply by (x-2)²
1 < 9(x-2)²
(3x-7)(3x-5) > 0 and so on.

Now you have altered the original inequality, which then produce different values when you are working it out, thus you get the wrong answer.

 

[Riviet]

You can also show this graphically, by sketching the graph of y=1/(x-2) and y=1/(x-2)² and shading the respective areas under each.

 

[NickP101]

Ok so if i had the equation y = 3x + 5 i can square both sides and keep the eqn =. But with inequalities u cant do that, is that correct

So if i had Sqrt(x) < 2, i cant square both sides to get x < 4?

Thanks for the help guys!

 

[_ShiFTy_]

Ok so if i had the equation y = 3x + 5 i can square both sides and keep the eqn =. But with inequalities u cant do that, is that correct

If you sqare that equation, to sketch it, you would have to draw y = 3x + 5 and y = -3x - 5, because of the +-

 

[Riviet]

So if i had Sqrt(x) < 2, i cant square both sides to get x < 4?

Thanks for the help guys!

By squaring both sides, you have added in negative solutions of x<4, which give unreal roots if you substitute these x values into sqrt(x) < 2. So therefore you can't just square both sides in this case.

 

[Slidey]

Applying monotonic functions to an inequality won't affect its basic nature in any way (though you must of course considering whether it is increasing or decreasing).

y=x^2 is not a monotonic functions, thus it cannot be applied without losing information.

 

[yinyin]

what the...
i just draw a number line and graphic it....
ive got bigger than 7/3 smaller or equal to 2.
is this the right answer?