The problem is to solve the inequality $(\frac{1}{2})^x < 8$ for $x$.

1. Problem Description

The problem is to solve the inequality $(\frac{1}{2})^x < 8$ for $x$.

2. Solution Steps

First, rewrite the inequality in terms of powers of

2. $(\frac{1}{2})^x < 8$

We can rewrite $\frac{1}{2}$ as $2^{-1}$ and $8$ as $2^3$. So, we have:

$(2^{-1})^x < 2^3$

Using the power of a power rule, we get:

$2^{-x} < 2^3$

Since the base is greater than 1, we can compare the exponents while preserving the inequality:

$-x < 3$

Multiply both sides by -

1. Remember to reverse the inequality sign when multiplying by a negative number:

$x > -3$

3. Final Answer

$x > -3$