Exponential Inequalities  TOPIC_SOLVED

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Exponential Inequalities

Postby Hal on Tue Mar 18, 2014 9:05 pm

I'm having trouble solving my homework and I would like some help.

S = [ -2 ; 2 ]. Check if the following sentence is true or false:

a) 2^(2x) - 2^x <= 0, for every X that belongs to S.

I can see that this is false because if I substitute X for 2 it is not <= 0, but i wanted to prove that the sentence was false.

What I did: 2^x = t.

t^2 - t <= 0 and found that the values i'm looking for are between 0 and 1

First root: 2^x <= 1
x <= 0

Second root: 2^x >= 0, so any value i can give to X will turn this sentence >= 0. And, of course, these values belong to S -> the sentence is true according to what I did, but my book says it is false.

Sorry for any mistake, english is not my native language and some notations might be wrong (I don't know if it is correct to call that a ''sentence'').



-
Hal
 
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Re: Exponential Inequalities

Postby theshadow on Wed Mar 19, 2014 11:45 am

Hal wrote:S = [ -2 ; 2 ]. Check if the following sentence is true or false:

a) 2^(2x) - 2^x <= 0, for every X that belongs to S.

I can see that this is false because if I substitute X for 2 it is not <= 0, but i wanted to prove that the sentence was false.

I'm not sure what you mean here? If you're saying that you found an x inside S where the sentence is false, then that IS the proof that it's false. You prove it's false by a counterexample. So you're done as soon as you find ANY x inside S where the sentence is false.
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Re: Exponential Inequalities

Postby Hal on Wed Mar 19, 2014 12:56 pm

I meant I wanted to prove that the sentence was false using the exponential inequalities ''step by step''. There will be exercises that I can't just substitue X for a number because the sentence is too big or something else.

Sorry if it wasn't clear,
Hal
 
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Re: Exponential Inequalities  TOPIC_SOLVED

Postby nona.m.nona on Wed Mar 19, 2014 1:44 pm

In such a case, use the properties of the given expression. Applying logic to the left-hand side, one obtains:

2^(2x) - 2^x = (2^x)^2 - 2^x = 2^x (2^x - 1)

Since the base is positive, 2^x will always be positive. Since 2^x = 1 for x = 0 and since 2 > 1 so 2^x is growing exponentially for x > 0, then 2^x - 1 > 0 for x > 0. As such, the product of these two non-negative factors must be non-negative on [0, +infinity). Thus the claimed inequality is shown to be false on the stated interval.
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Re: Exponential Inequalities

Postby Hal on Wed Mar 19, 2014 2:20 pm

Problem solved. Thanks for the answer!
Hal
 
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Joined: Tue Mar 18, 2014 8:23 pm


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