4.2.4 Use your graph to determine the values of \( x \) for which \( \frac{4}{x-1} \geq-2 \)

To solve \( \frac{4}{x-1} \geq -2 \), first, you can rewrite the inequality as \( \frac{4}{x-1} + 2 \geq 0 \). This leads to \( \frac{4 + 2(x - 1)}{x - 1} \geq 0 \), simplifying to \( \frac{2x + 2}{x - 1} \geq 0 \) or \( \frac{2(x + 1)}{x - 1} \geq 0 \). The critical points are when the numerator is zero (at \( x = -1 \)) and when the denominator is zero (at \( x = 1 \)). Analyzing the signs in the intervals split by these points will help you determine the values of \( x \). This inequality holds true in the intervals \( x \in (-\infty, -1] \) and \( x \in (1, \infty) \), with the important note that \( x = 1 \) is excluded since it makes the denominator zero. Thus, your final solution for \( x \) is \( (-\infty, -1] \cup (1, \infty) \). Enjoy graphing it!