[STPM] Maths T needs help!!

[mitsukolyn]

I need help on this question...

Question:

Find the set of values of x for the following inequalities.
b)e^x - 3e^-x ? 2 [hint: multiply both sides by e^x]

[ken_cck_kid]

e^x - 3e^-x ? 2
e^x << 2 + 3e^-x
e^2x << 2e^x + 3 ( multiply by e^x)
e^2x-2e^x-3<<0 (a quadratic equation)
(e^x-3)(e^x+1) <<0

try to continue ya, tis should be the way la if i'm not mistaken..haha..

[luke]

e^x - 3e^(-x) < 2
e^(2x) - 3 < 2e^x
e^(2x) - 2e^x - 3 < 0

let Z = e^x, therefore: Z^2 - 2Z - 3 < 0 (voila! normal quadratic inequality!)

after a bit of drawing the curve we figure out that, -1 < Z < 3

substitute back, -1 < e^x < 3
but e^x cannot be less than 0, thus 0 =< e^x < 3

ln 0 =< ln (e^x) < ln 3

negative infinity =< x < ln 3

simply means x < ln 3

[mitsukolyn]

Originally Posted by luke

e^x - 3e^(-x) < 2
e^(2x) - 3 < 2e^x
e^(2x) - 2e^x - 3 < 0

let Z = e^x, therefore: Z^2 - 2Z - 3 < 0 (voila! normal quadratic inequality!)

after a bit of drawing the curve we figure out that, -1 < Z < 3

substitute back, -1 < e^x < 3
but e^x cannot be less than 0, thus 0 =< e^x < 3

ln 0 =< ln (e^x) < ln 3

negative infinity =< x < ln 3

simply means x < ln 3

haha..im stuck in the half way thts y i turn to here...wow...ur so smart!!! but cn i knw wad's ln means and when to use it??

[chiunlin]

ln = log to the base e/natural log

[mitsukolyn]

Originally Posted by chiunlin

ln = log to the base e/natural log

thanks!! i got it d...

[lonewolf5999]

As in the hint, multiply both sides to obtain

e^2x - 3 < 2e^x

use the substitution u = e^x to get

u^2 -3 < 2u, which is easily solved

substitute back e^x, but note that e^x is always positive, so you will need to reject any answers contradicting this fact.

[tomatoinc]

luke is still da smart ass.

Originally Posted by luke

e^x - 3e^(-x) < 2
e^(2x) - 3 < 2e^x
e^(2x) - 2e^x - 3 < 0

let Z = e^x, therefore: Z^2 - 2Z - 3 < 0 (voila! normal quadratic inequality!)

after a bit of drawing the curve we figure out that, -1 < Z < 3

substitute back, -1 < e^x < 3
but e^x cannot be less than 0, thus 0 =< e^x < 3

ln 0 =< ln (e^x) < ln 3

negative infinity =< x < ln 3

simply means x < ln 3

[Lance2]

Nice post, Keep up the good work man.

[smallkid]

I found this question in fajar bakti ace ahead (volume 2)
Exam practise chapter 8
Q21.

Suppose the time T taken for a car wash is normally distributed with mean 6minutes and standard deviation 1.5 minutes.

c) If 1000 cars are washed, what is the probability that not more than one car takes more than 10.62 minutes.

Answer : 0.3399

Anyone knows how to do it?

I've tried poisson approximation, but i'm still no where near the solution