List the critical values of the related function. Then solve the inequality. \[ \frac{4 x}{x+6} \geq 0 \] The critical value(s) is/are (Simplify your answer. Type an integer or a simplified fraction. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)
Solution
Step 1 ：The critical values of a function are the values of x that make the function equal to zero or undefined. In this case, the function is undefined when the denominator is zero, i.e., \(x = -6\). The function is zero when the numerator is zero, i.e., \(x = 0\). So, the critical values are \(x = -6\) and \(x = 0\).
Step 2 ：To solve the inequality, we need to consider the intervals determined by the critical values and test a value in each interval. The intervals are \((-\infty, -6)\), \((-6, 0)\), and \((0, \infty)\).
Step 3 ：The critical values are \(x = 0\) and \(x = -6\). The inequality holds for the intervals \((-\infty, -6)\) and \((0, \infty)\).
Step 4 ：Final Answer: The critical values are \(\boxed{-6, 0}\). The solution to the inequality is \(\boxed{x < -6, x > 0}\).
