# Inequalities, trigonometric and x exponent.

1. Oct 30, 2013

### Mutaja

These are the two last problems I'll bother you with for a short while (I love this forum, I'll definitely stay on and hopefully be able to contribute in the future).

1. The problem statement, all variables and given/known data
Problem 1:
($-x^2$-1)sin2x > 0 , xe[0,2$\pi$]

Problem 2:
$2^{-x^2+x+2}$ < 4

2. Relevant equations

3. The attempt at a solution

For the first problem:

($-x^2$-1)sin2x > 0 , xe[0,2$\pi$]

Since x is an element within a positive boundary, can I write the problem as:

sin(2x) ≤ 0

Why, or why not?

Other than that, I need to refresh my knowledge on solving inequalities.

Any help or input is appreciated while I attempt to gather more info.

2. Oct 30, 2013

### Staff: Mentor

Your first inequality can be rewritten as -(x2 + 1)sin(2x) ≥ 0, or
(x2 + 1)sin(2x) ≤ 0

Since x2 + 1 ≥ 1 for all real x, it can never be zero or negative. If you divide both sides of the inequality by x2 + 1, and the direction of the inequality won't change. That gets you to the inequality you wrote.

BTW, your mix of LaTeX and HTML tags threw me off for a bit. In LaTeX use leq for ≤. Using the U (underscore) HTML tag under a < symbol confused me for a bit.

You can also click Go Advanced to see a table of quick symbols off to the right. It has Greek letters and symbols such as ≤, ≥, ≠, ∞, and several others.

3. Oct 30, 2013

### Mutaja

But how do I go from sin2x ≤ 0 to something like x = $\pi$ + n $\pi$/2? This is just something random I wrote down, but hopefully you get the idea.

Where do I even look to find the solution for this?

Also, in respect to my 2nd inequality. I don't know where I should look for tips on how to solve it. My book says absolutely nothing about inequality in this regard. It might sound lazy, but I've looked. I suspect I'm supposed to remember this from pre-calculus math but it's as unfortunate as it's obvious - I don't.

Sorry I'm not able to make more progress before posting again. Again, thanks for all your help and any input is greatly appreciated.
I'm sorry, I didn't know it mattered as I can't see any visual difference, but I'll keep that in mind. Thanks.

4. Oct 30, 2013

### Staff: Mentor

You're not going to get x = ... as the solution to sin(2x) ≤ 0. The simplest way is to sketch a quick graph of y = sin(2x) and note the intervals where the graph touches or goes below the x-axis.

For your second problem, which is
$$2^{-x^2 + x + 2} < 4$$

Note the 4 = 22, so then you have
$$2^{-x^2 + x + 2} < 2^2$$

Since y = 2x is a strictly increasing function, if 2A < 2B, then A < B.

5. Oct 30, 2013

### Staff: Mentor

This was in reference to what I said about using the canned symbols ≤ rather than [ U]<[/ U]; i.e., coming up with your own way to write ≤. A lot of us here at PF will "quote" your post, which means that we're looking at the LaTeX and/or HTML markup, instead of how the post actually appears in the browser, so your ≤ actually seemed to me at first to be <, which was why I was confused.