Understanding the Problem
To find the solution set for the inequalities involving polynomials, we first need to analyze each expression and determine when they are less than or greater than zero. We will tackle each part of the equation separately.
Breaking Down the Inequalities
1. First Inequality: 0 < 35x4 + 8x5 + 2
We aim to solve the inequality:
35x4 + 8x5 + 2 > 0
Since both the terms with positive powers of x are always non-negative for real x, and since there is a constant term of +2, the entire expression is positive for all real values of x. Hence, this inequality has no restrictions on x.
2. Second Inequality: 9x8 + 8x8 + 8 < 0
This inequality simplifies to:
17x8 + 8 < 0
Here, both terms are non-negative for all real x since x8 is always non-negative and adding 8 makes it strictly positive. Therefore, this inequality provides no solutions in the real number system.
Combining the Results
Since the first inequality is valid for all real numbers while the second inequality does not provide any valid solutions, the overall solution set would be empty when considering both inequalities simultaneously.
Conclusion
Thus, the solution set to the given inequalities is:
∅