Find all r ∈ R such that r<1+1/r?
Find all r ∈ R such that r<1+1/r?
So I'm having difficulty with this problem.
I've figured out that r is a variable and r does not = 0 and by solving it, multiplying both sides by r is incorrect.
Can you guys help me with this problem?
My first instinct is to just multiply both sides by r but that's apparently wrong.
Any help is appreciated.
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The problem with multiplying by r is that you don't know its sign---remember that inequalities change direction when multiplying by negative numbers.
r < 1 + 1/r ==> r - 1/r - 1 < 0.
Get a common denominator on the left---this doesn't require you multiply through so it is safe.
r²/r - 1/r - r/r < 0 ==> (r² - r - 1)/r < 0.
The quotient is negative if the numerator and denominator have opposite signs. Find where they are zero.
r² - r - 1 = 0 ==> r = 1/2 ± √(5)/2, and r = 0 if r = 0.
Check the three intervals
-∞ < r < (1 - √(5))/2,
(1 - √(5))/2 < r < 0,
0 < r < (1 + √(5))/2, and
(1 + √(5))/2 < r < ∞.
The expression (r² - r - 1)/r is negative on the first and third interval listed above. This is your solution set.
-∞ < r < (1 - √(5))/2, and 0 < r < (1 + √(5))/2.
(Try to avoid multiplication and division when solving inequalities. Addition and subtraction are safer.)
r < 1 + 1/r
r < r/r + 1/r
r < (r + 1)/r
For r < 0, multiplying both sides by r yields
r^2 > r + 1
r^2 - r > 1
r^2 - r + 1/4 > 1 + 1/4
(r - 1/2)^2 > 5/4
r - 1/2 < -â5/2 or r > â5/2
r < -â5/2 + 1/2 or r > â5/2 + 1/2
r < 0 was specified so this becomes
r < -â5/2 + 1/2
For r > 0, multiplying both sides by r yields
r^2 < r + 1
r^2 - r < 1
r^2 - r + 1/4 < 1 + 1/4
(r - 1/2)^2 < 5/4
-â5/2 < r - 1/2 < â5/2
-â5/2 + 1/2 < r < â5/2 + 1/2
r > 0 was specified so this becomes
0 < r < â5/2 + 1/2
Therefore, the solution is the union of these two intervals: {r| r < -â5/2 + 1/2 or 0 < r < â5/2 + 1/2}