1. **Stating the problem:** We are working with inequalities that describe real-world situations, such as a submarine descending, water rising in a lake, and solving algebraic inequalities. 2. **Important rules:** - Inequality symbols: $>$ (greater than), $\geq$ (greater than or equal to), $<$ (less than), $\leq$ (less than or equal to). - When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality symbol. 3. **Submarine problem:** - Given rate of change: $-10$ feet per second. - Inequality for depth $-140$ feet or deeper: $$-10t \leq -140$$ 4. **Solving the submarine inequality:** - Divide both sides by $-10$ (negative number), reverse inequality: $$\cancel{-10}t \geq \cancel{-10} \frac{-140}{-10}$$ $$t \geq 14$$ - This means the submarine takes at least 14 seconds to reach $-140$ feet or deeper. 5. **Values that satisfy the inequality:** - $t=14, 15, 20$ all satisfy $t \geq 14$. 6. **Lake water problem:** - Water level plus rise $r$ must not exceed dam height 152: $$127 + r \leq 152$$ 7. **Solving for $r$:** - Subtract 127 from both sides: $$\cancel{127} + r \leq 152 - 127$$ $$r \leq 25$$ - Values like $r=0, 5, 25$ satisfy this. 8. **Inequality solving examples:** A. $m - 0.3 > 1.45$ - Add 0.3: $$m > 1.45 + 0.3$$ $$m > 1.75$$ B. $3.5n \leq 7$ - Divide by 3.5 (positive): $$n \leq \frac{7}{3.5}$$ $$n \leq 2$$ C. $-2 \geq b + \frac{1}{8}$ - Subtract $\frac{1}{8}$: $$-2 - \frac{1}{8} \geq b$$ $$-\frac{17}{8} \geq b$$ - Rewrite: $$b \leq -\frac{17}{8}$$ D. $-\frac{1}{2}y < 6$ - Multiply both sides by $-2$ (negative, reverse inequality): $$y > -12$$ E. $-x < -4$ - Multiply both sides by $-1$ (negative, reverse inequality): $$x > 4$$ F. $-\frac{2a}{3} > 6$ - Multiply both sides by $-\frac{3}{2}$ (negative, reverse inequality): $$a < -9$$ **Summary:** - Always reverse inequality when multiplying/dividing by negative. - Check solutions by substitution. - Inequalities describe ranges of values, not just single numbers. Final answers: - Submarine: $t \geq 14$ - Lake: $r \leq 25$ - Inequalities A-F as above.