1. **State the problem:** Solve the inequality $\frac{4}{x} \geq x$. 2. **Rewrite the inequality:** Multiply both sides by $x$, but remember to consider the sign of $x$ because it affects the inequality direction. 3. **Case 1: $x > 0$** Multiply both sides by $x$ (positive, so inequality direction stays the same): $$4 \geq x^2$$ This means: $$x^2 \leq 4$$ Taking square roots: $$-2 \leq x \leq 2$$ Since $x > 0$, the solution here is: $$0 < x \leq 2$$ 4. **Case 2: $x < 0$** Multiply both sides by $x$ (negative, so inequality direction reverses): $$4 \leq x^2$$ This means: $$x^2 \geq 4$$ Taking square roots: $$x \leq -2 \quad \text{or} \quad x \geq 2$$ Since $x < 0$, the solution here is: $$x \leq -2$$ 5. **Combine solutions and exclude $x=0$ (undefined):** $$x \leq -2 \quad \text{or} \quad 0 < x \leq 2$$ **Final answer:** $$\boxed{x \in (-\infty, -2] \cup (0, 2]}$$