Harder Inequations Question (1 Viewer)

[-Dark Light-]

This is from Fitzpatrick 3U page 60.

Find the values of x for which the following inequations are simultaneously satisfied:



Full worked solution will be great because this one is a tough one.

Thanks

 

[Trebla]

I'm assuming you mean both expressions are negative?

 

[-Dark Light-]

I'm assuming you mean both expressions are negative?

I edited it. One is positive.

 

[Trebla]

This is from Fitzpatrick 3U page 60.

Find the values of x for which the following inequations are simultaneously satisfied:



Full worked solution will be great because this one is a tough one.

Thanks

The second inequality gives -1 < x < 2 through the usual approach

The first inequality can be broken up into two cases (note x needs to be non-zero):
Case 1: Consider x > 0
x + 1/x > 0
=> x(x² + 1) > 0
=> x > 0 since x² + 1 > 0 for all real x
i.e. The solution is the entire set of x > 0 in accordance to the case we are considering

Case 2: Consider x < 0
x - 1/x > 0
=> x(x² - 1) > 0
=> - 1 < x < 0 and x > 1 which can be easily found using a graphical approach
However, we have restricted attention to the set x < 0 so we only take - 1 < x < 0

Hence the solution to x + 1/|x| > 0 is
- 1 < x < 0 and x > 0
Intersect this with the set -1 < x < 2 from the second inequality gives
- 1 < x < 0 and 0 < x < 2

 

[-Dark Light-]

The second inequality gives -1 < x < 2 through the usual approach

The first inequality can be broken up into two cases (note x needs to be non-zero):
Case 1: Consider x > 0
x + 1/x > 0
=> x(x² + 1) > 0
=> x > 0 since x² + 1 > 0 for all real x
i.e. The solution is the entire set of x > 0 in accordance to the case we are considering

Case 2: Consider x < 0
x - 1/x > 0
=> x(x² - 1) > 0
=> - 1 < x < 0 and x > 1 which can be easily found using a graphical approach
However, we have restricted attention to the set x < 0 so we only take - 1 < x < 0

Hence the solution to x + 1/|x| > 0 is
- 1 < x < 0 and x > 0
Intersect this with the set -1 < x < 2 from the second inequality gives
- 1 < x < 0 and 0 < x < 2

Thanks for the quick reponse dude!