## Exponential Inequalities

Quadratic equations and inequalities, variation equations, function notation, systems of equations, etc.

### Exponential Inequalities

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

Posts: 3
Joined: Tue Mar 18, 2014 8:23 pm

### Re: Exponential Inequalities

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.

Posts: 82
Joined: Sun Feb 22, 2009 11:12 pm

### Re: Exponential Inequalities

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

Posts: 3
Joined: Tue Mar 18, 2014 8:23 pm

### Re: Exponential Inequalities

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.
nona.m.nona

Posts: 249
Joined: Sun Dec 14, 2008 11:07 pm

### Re: Exponential Inequalities

Problem solved. Thanks for the answer!
Hal

Posts: 3
Joined: Tue Mar 18, 2014 8:23 pm