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How to combine inequalities

How would I combine -2<x-1<0 and 1>(1/x+2)>1/3 to get Mod(x-1/x+2) <2? Is there a general rule on how to do this? )
Reply 1
Avatar for Zacken
Zacken
22
We have:

.

And

.

Can you take it from here?
Reply 2
Avatar for Lunu
Lunu
OP
1
Thank you!!!
Original post by Lunu
How would I combine -2<x-1<0 and 1>(1/x+2)>1/3 to get Mod(x-1/x+2) <2? Is there a general rule on how to do this? )


1. Latex is your friend.

2. I would write:

−2<x−1<0⇒0<1−x<2-2 < x-1 < 0 \Rightarrow 0 < 1-x < 2

and then since all quantities are non-negative in both inequalities, we can multiply them to get:

0<1−xx+2<2⇒∣1−xx+2∣<2⇒∣x−1x+2∣<20 < \frac{1-x}{x+2} < 2 \Rightarrow |\frac{1-x}{x+2}| < 2 \Rightarrow |\frac{x-1}{x+2}| < 2

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