on list tion 37 stion 38 stion 39 stion 40 Solve the inequality. \[ x^{2} \geq 8 x \] Choose the correct answer below. A. \( \square \) \( \) \( \square \) C. \( x \leq \) \( \square \) or \( x \geq \) \( \square \) D. \( \square \) sxs \( \square \) E. No solution

To solve the inequality \(x^{2} \geq 8x\), first rearrange it into standard form: \(x^{2} - 8x \geq 0\). Next, factor the left-hand side: \(x(x - 8) \geq 0\). Now, find the critical points by setting the expression to zero: \(x = 0\) and \(x = 8\). To determine where the product is non-negative, test intervals: 1. For \(x < 0\) (e.g., \(-1\)): \((-)(-)\) = positive. 2. For \(0 < x < 8\) (e.g., \(4\)): \(+(−)\) = negative. 3. For \(x > 8\) (e.g., \(9\)): \(+(+)\) = positive. The solution is where the product is zero or positive, i.e., \(x \leq 0\) or \(x \geq 8\). Thus, the correct answer is **C: \( x \leq 0 \) or \( x \geq 8 \)**. --- To clarify, this inequality shows the behavior of parabolas, which are defined by quadratic functions. Understanding how to solve them is vital in algebra and calculus since they appear frequently in real-world situations, such as optimizing areas or projectile motion, where you might want to know at which points the path remains above or below a certain height. In learning to solve inequalities like these, a common mistake is forgetting to test the intervals between critical points. It's not enough to find the zeros; you have to check how the function behaves in the intervals created by these points. The sign of the intervals tells you where the inequality holds, guiding you to the correct solution.