Hi all, I need your help to solve an inequality involving absolute values.
I would like to know if (i) what I did is correct or not and (ii) if not do you have any intuitions to simplify or solve this inequality ?
[MATH] (C_H - C_L) (\lvert \bar x - v + P_{L} \rvert - \bar x + P_{L}) > \gamma [/MATH]
Here is what I did so far :
[MATH] (C_H - C_L) (\lvert \bar x - v + P_{L} \rvert) +(C_H - C_L)(- \bar x + P_{L}) > \gamma \\ (\lvert \bar x - v + P_{L} \rvert) - (\bar x + P_{L}) > \frac{\gamma}{(C_H - C_L)} \\ (\lvert \bar x - v + P_{L} \rvert) > \frac{\gamma}{(C_H - C_L)} + \bar x - P_{L}[/MATH]
So we have to solve:
[MATH](1): ( \bar x - v + P_{L} ) > \frac{\gamma}{(C_H - C_L)} + \bar x - P_{L}[/MATH]
and
[MATH](2): ( \bar x - v + P_{L} ) > -\frac{ \gamma}{(C_H - C_L)} - \bar x + P_{L}[/MATH]
This would mean that :
[MATH](1): \frac{-v+2P_L}{C_H-C_L} > \gamma\\ (2): - \frac{2\bar{x}-v}{C_H-C_L} > \gamma[/MATH]
Is it correct ?
Thanks in advance, Marc
I would like to know if (i) what I did is correct or not and (ii) if not do you have any intuitions to simplify or solve this inequality ?
[MATH] (C_H - C_L) (\lvert \bar x - v + P_{L} \rvert - \bar x + P_{L}) > \gamma [/MATH]
Here is what I did so far :
[MATH] (C_H - C_L) (\lvert \bar x - v + P_{L} \rvert) +(C_H - C_L)(- \bar x + P_{L}) > \gamma \\ (\lvert \bar x - v + P_{L} \rvert) - (\bar x + P_{L}) > \frac{\gamma}{(C_H - C_L)} \\ (\lvert \bar x - v + P_{L} \rvert) > \frac{\gamma}{(C_H - C_L)} + \bar x - P_{L}[/MATH]
So we have to solve:
[MATH](1): ( \bar x - v + P_{L} ) > \frac{\gamma}{(C_H - C_L)} + \bar x - P_{L}[/MATH]
and
[MATH](2): ( \bar x - v + P_{L} ) > -\frac{ \gamma}{(C_H - C_L)} - \bar x + P_{L}[/MATH]
This would mean that :
[MATH](1): \frac{-v+2P_L}{C_H-C_L} > \gamma\\ (2): - \frac{2\bar{x}-v}{C_H-C_L} > \gamma[/MATH]
Is it correct ?
Thanks in advance, Marc