To solve the inequality [tex]\( x^7 + x^2 \geq 5x^5 + 5 \)[/tex], let's go through the process step-by-step:
1. Rewrite the inequality:
[tex]\[
x^7 + x^2 - 5x^5 - 5 \geq 0
\][/tex]
We need to solve for [tex]\( x \)[/tex] such that the expression on the left side is greater than or equal to zero.
2. Analyze the expression:
We observe that the inequality is defined by a polynomial function [tex]\( f(x) = x^7 + x^2 - 5x^5 - 5 \)[/tex].
3. Identifying critical points:
To solve this, we need to find the roots of the equation [tex]\( x^7 + x^2 - 5x^5 - 5 = 0 \)[/tex] as these points will help determine where the inequality changes its sign.
4. Solve the inequality:
By analyzing the roots and behavior of the polynomial function [tex]\( f(x) \)[/tex] within its defined intervals, we determine where the given inequality holds true.
The solution set for the inequality [tex]\( x^7 + x^2 \geq 5x^5 + 5 \)[/tex] is found to be:
[tex]\[ \left((-\sqrt{5} \leq x \leq -1)\right) \cup \left(\sqrt{5} \leq x < \infty \right) \][/tex]
Therefore, the solution set is:
[tex]\[ ((x \leq -1) \, \text{and} \, (-\sqrt{5} \leq x)) \cup ((\sqrt{5} \leq x) \, \text{and} \, (x < \infty)) \][/tex]
So, rewritten more concisely, the solution set is:
[tex]\[ ((x \leq -1) \, \text{and} \, (-\sqrt{5} \leq x)) \cup ((\sqrt{5} \leq x) \, \text{and} \, (x < \infty)) \][/tex]
This means that the values of [tex]\( x \)[/tex] satisfying the inequality are those in the intervals [tex]\([-1, -\sqrt{5}]\)[/tex] and [tex]\([\sqrt{5}, +\infty)\)[/tex].
