How Do You Solve the Inequality \(\frac{1}{2^x} > \frac{1}{x^2}\)?

[zeion]

Homework Statement

Use your knowledge of exponents to solve

$$\frac{1}{2^x} > \frac{1}{x^2}$$

Homework Equations

The Attempt at a Solution

$$x^2 > 2^x$$

Then I am stuck.

I know they intersect at x = 2.

 

[micromass]

Try to find out when the equality holds. Let's say they hold at a and b. This will give you regions $$]-\infty,a[, ]a,b[, ]b,+\infty[$$. From any region, pick a point and check if the inequality is satisfied at that point. If so, then that region is part of the solution.

 

[zeion]

Try to find out when the equality holds. Let's say they hold at a and b. This will give you regions $$]-\infty,a[, ]a,b[, ]b,+\infty[$$. From any region, pick a point and check if the inequality is satisfied at that point. If so, then that region is part of the solution.

Thing is I don't know how to find what a and b are. (Except for 2)

 

[micromass]

Well, yeah, I don't think you can explicitly find the value of a and b (except for 2). But you know from the graph that such an a and b exist and where they lie.

 

[zgozvrm]

The Attempt at a Solution

$$x^2 > 2^x$$

Then I am stuck.

I know they intersect at x = 2.

They intersect at x=2 because [itex]2^2 = 2^2[/tex]

What about x=4?
Does [itex]4^2 = 2^4[/tex] ?

 

[micromass]

Yes, but how do you find the negative intersection?

 

[Mentallic]

You approximate. And if you're in a higher college math class, you find it in terms of the Lambert W function. Either way, you can't express the root of x²=2^x, x<0 explicitly in terms of elementary functions.