1. **State the problem:** Solve the inequality $x + 5x \geq 0$ and find the critical points from the equations $x - 3 = 0$ and $x + 5 = 0$. 2. **Simplify the inequality:** $$x + 5x \geq 0$$ Combine like terms: $$6x \geq 0$$ 3. **Solve the inequality:** Divide both sides by 6 (positive number, so inequality direction stays the same): $$\cancel{6}x \geq \cancel{6}0 \implies x \geq 0$$ 4. **Find critical points from equations:** From $x - 3 = 0$, solve for $x$: $$x = 3$$ From $x + 5 = 0$, solve for $x$: $$x = -5$$ 5. **Interpret the number line:** The critical points are $-5$ and $3$. The inequality solution is $x \geq 0$. 6. **Final answer:** The solution to the inequality is: $$x \geq 0$$ The critical points $x = -5$ and $x = 3$ are marked on the number line as reference points.