1. **Understanding Inequalities:** An inequality is a mathematical sentence that shows the relationship between two quantities that are not necessarily equal. The symbols used include $>$ (greater than), $<$ (less than), $\geq$ (greater than or equal to), and $\leq$ (less than or equal to). 2. **Writing Inequalities from Situations:** For example, a submarine descending at a rate of $-10$ feet per second reaching $-140$ feet or deeper can be represented as: $$-10t \leq -140$$ where $t$ is time in seconds. 3. **Solving Inequalities:** To solve $-10t \leq -140$, divide both sides by $-10$. Remember, dividing by a negative number reverses the inequality symbol: $$\cancel{-10}t \geq \cancel{-10}14$$ $$t \geq 14$$ This means the submarine takes at least 14 seconds to reach $-140$ feet or deeper. 4. **Checking Solutions:** Substitute values like $t=14, 15, 20$ into the original inequality to verify they satisfy it. 5. **Graphing Inequalities:** The solution $t \geq 14$ is graphed on a number line with a closed circle at 14 and shading to the right, indicating all values greater than or equal to 14. 6. **Writing Inequalities for Real-World Problems:** For example, if a lake's water level is $127$ feet and can rise up to $152$ feet before overflowing, the inequality is: $$127 + r \leq 152$$ where $r$ is the rise in water level. 7. **Solving and Interpreting Inequalities:** Solve for $r$: $$r \leq 25$$ meaning the water can rise up to 25 feet without overflowing. 8. **Applying Properties of Inequalities:** When multiplying or dividing both sides by a negative number, reverse the inequality symbol. For example: $$-0.3m > 1.45 \Rightarrow m < -\frac{1.45}{0.3} = -4.83$$ 9. **Solving One-Step Inequalities:** Examples include: - $3.5n \leq 7 \Rightarrow n \leq 2$ - $-x < -4 \Rightarrow x > 4$ 10. **Solving Two-Step Inequalities:** For example, to solve $3x + 5 \leq 25$: - Subtract 5: $3x \leq 20$ - Divide by 3: $x \leq \frac{20}{3} \approx 6.67$ 11. **Writing Inequalities from Word Problems:** For example, if Caitlyn has $25$ to spend, must pay back $5$, and sheets cost $3$ each, the inequality is: $$3x + 5 \leq 25$$ where $x$ is the number of sheets. 12. **Solving and Interpreting Solutions:** Solve for $x$: $$3x \leq 20 \Rightarrow x \leq \frac{20}{3} \approx 6.67$$ So Caitlyn can buy up to 6 sheets. 13. **Graphing Solutions:** Use number lines with open or closed circles to represent inequalities, shading the solution region. 14. **Summary:** Inequalities model real-world constraints and conditions. Solving them involves isolating the variable while respecting inequality rules, especially reversing the inequality when multiplying or dividing by negatives. Graphing helps visualize solution sets. **Final Note:** Always check solutions by substitution and interpret them in the problem's context.