\( y ^ { 3 } x \geq y ^ { - 2 } \)

To solve the inequality \( y^{3} x \geq y^{-2} \), we can first rearrange it to bring all terms to one side: \( y^{3} x - y^{-2} \geq 0 \). To eliminate the fraction, we can multiply through by \( y^{2} \) (assuming \( y \neq 0 \)), resulting in \( y^{5} x - 1 \geq 0 \). This leads us to \( y^{5} x \geq 1 \), indicating that for positive \( x \), we need \( y \) to satisfy \( y \geq (1/x)^{1/5} \). It’s always useful to remember that inequalities can change when multiplying or dividing by negative values; hence, the signs of \( x \) and \( y \) play a crucial role in determining the solution set. To visualize this, considering graphs can be really effective. Plotting \( y^{5} x = 1 \) can show how the inequality behaves across different quadrants, helping you to understand where \( y \) is constrained by the value of \( x \) in a tangible way.