1. **Understanding the problem:** We are given inequalities related to real-world situations: a submarine descending and a lake water level rising. 2. **Key concepts:** An inequality shows that two quantities are not equal but related by symbols like $<$, $\leq$, $>$, $\geq$. 3. **Submarine problem:** - The submarine descends at a rate of $-10$ feet per second. - We want to find the time $t$ when the submarine reaches $-140$ feet or deeper. 4. **Write the inequality:** $$-10t \leq -140$$ 5. **Solve the inequality:** - Divide both sides by $-10$ (remember to reverse the inequality symbol because we divide by a negative): $$\cancel{-10}t \geq \cancel{-10}14$$ $$t \geq 14$$ 6. **Interpretation:** - The submarine takes at least 14 seconds to reach $-140$ feet or deeper. - Values like $t=14, 15, 20$ satisfy the inequality. 7. **Lake water level problem:** - Water level rises from 127 ft toward the dam top at 152 ft. - Let $r$ be the rise per hour. 8. **Write the inequality:** $$127 + r \leq 152$$ 9. **Solve the inequality:** $$r \leq 152 - 127$$ $$r \leq 25$$ 10. **Interpretation:** - The water can rise up to 25 ft per hour without overflowing. - Values like $r=0, 1, 5$ satisfy the inequality. 11. **Properties of inequalities:** - When multiplying or dividing by a negative number, reverse the inequality symbol. - Otherwise, solve like equations. 12. **Additional practice problems:** A. $-0.3m > 1.45$ solve for $m$: $$m < \frac{1.45}{-0.3}$$ $$m < -4.8333$$ B. $3.5n \leq 7$ solve for $n$: $$n \leq 2$$ C. $-2 \geq b + \frac{1}{6}$ solve for $b$: $$b \leq -2 - \frac{1}{6} = -\frac{13}{6}$$ D. $-\frac{1}{2}y < 6$ solve for $y$: $$y > -12$$ E. $-x < -4$ solve for $x$: $$x > 4$$ F. $-\frac{2a}{3} > 6$ solve for $a$: $$a < -9$$ **Summary:** Inequalities show ranges of values that satisfy conditions. When dividing or multiplying by negatives, reverse the inequality sign. Substitute values to check if they satisfy the inequality.