SOLUTION: I am sending two questions, because I think they are very similar, although I was able to solve only the first one, but in a somewhat unsatisfactory way: 1) {{{ (1/log(2,x))<=(

Algebra ->  Algebra  -> Logarithm Solvers, Trainers and Word Problems -> SOLUTION: I am sending two questions, because I think they are very similar, although I was able to solve only the first one, but in a somewhat unsatisfactory way: 1) {{{ (1/log(2,x))<=(      Log On

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Question 5387: I am sending two questions, because I think they are very similar, although I was able to solve only the first one, but in a somewhat unsatisfactory way:
1) +%281%2Flog%282%2Cx%29%29%3C=%281%2Flog%282%2Csqrt%28x%2B2%29%29%29+
2) +%281%2Flog%282%2Cx%29%29-%281%2F%28log%282%2Cx%29%29-1%29%3C1++
My goal would be to solve these inequalities purely by algebraic manipulation.
I examined both questions with Mathematica 4 and graphed them with Winplot (Peanut)and both confirm the answers, that by coincidence are the same for the two questions.
I would very much appreciate some help here.
Thanks
Andreas

Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
1%2F%28log%28x%29%29 <= 1%2F%28log%28sqrt%28x%2B2%29%29%29 -- cross multiply. Not sure if this changes the inequality round though!

%28log%28sqrt%28x%2B2%29%29%29 <= %28log%28x%29%29 -- raise everything to power 2.

sqrt%28x%2B2%29 <= x -- now square both sides

x%2B2 <= x%5E2

x%5E2+-+x+-+2 >= 0

(x+1)(x-2) >= 0

So, this is asking where is the quadratic greater or equal than zero ie greater or equal than the x-axis (y=0). Well, we know the roots are now x=-1 and x=+2, so the answer is x<=-1 or x>=2: assuming that the inequality doesn't swap round when we cross multiplies the logs in the first step.

Is that clear :-)?

jon.