Solve xx in the inequality

−x^10>−8⋅x^5


We start by analyzing the given inequality:

\[
- x^{10} > -8 \cdot x^5
\]

### Step 1: Eliminate the negative sign

To simplify the inequality, we can multiply both sides by \(-1\), which reverses the inequality:

\[
x^{10} < 8 \cdot x^5
\]

### Step 2: Divide both sides by \(x^5\)

Next, divide both sides by \(x^5\), assuming \(x \neq 0\) because division by zero is undefined:

\[
x^5 < 8
\]

### Step 3: Solve for \(x\)

Now, to solve for \(x\), we take the fifth root of both sides:

\[
x < \sqrt[5]{8}
\]

Since \(\sqrt[5]{8} = 2\), we have:

\[
x < 2
\]

### Step 4: Consider \(x = 0\)

However, when \(x = 0\), the original inequality becomes:

\[
0 > 0
\]

which is false, so \(x \neq 0\).

### Final Answer:

The solution set for the inequality is:

\[
x \in (-\infty, 2)
\]

### Do you want details or have any questions?

1. What happens if \(x = 2\) in the inequality?
2. Why do we reverse the inequality when multiplying by \(-1\)?
3. How do we find the fifth root of a number?
4. What are the implications of dividing by \(x^5\) for different values of \(x\)?
5. What if the inequality was \(x^{10} > 8x^5\)?
6. Can you generalize this solution method for other similar inequalities?
7. How does the inequality change if we have \(x^n < 8x^{m}\) where \(n > m\)?
8. What are the properties of roots and powers that we used here?

**Tip:** Always check for any restrictions, like \(x \neq 0\), when dividing both sides of an inequality by a variable term.

Learn how to solve the inequality -x^10 > -8*x^5 with detailed step-by-step instructions. This problem involves inequalities, exponents, and roots. Suitable for advanced high school students.

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