1. **Stating the problem:** We will explore how to solve inequalities and apply consumer arithmetic concepts such as discounts, taxes, and budgeting. 2. **Inequalities basics:** An inequality compares two expressions using symbols like $<$, $>$, $\leq$, or $\geq$. 3. **Rules for solving inequalities:** - You can add or subtract the same number on both sides without changing the inequality. - You can multiply or divide both sides by a positive number without changing the inequality. - If you multiply or divide both sides by a negative number, you must reverse the inequality sign. 4. **Example inequality:** Solve $3x - 5 < 10$. 5. **Step 1:** Add 5 to both sides: $$3x - 5 + 5 < 10 + 5$$ $$3x < 15$$ 6. **Step 2:** Divide both sides by 3 (positive number, so inequality stays the same): $$\frac{\cancel{3}x}{\cancel{3}} < \frac{15}{3}$$ $$x < 5$$ 7. **Consumer arithmetic example:** Suppose an item costs 120 and has a 20% discount. 8. **Step 1:** Calculate discount amount: $$120 \times 0.20 = 24$$ 9. **Step 2:** Subtract discount from original price: $$120 - 24 = 96$$ 10. **Step 3:** If a tax of 8% applies on the discounted price, calculate tax: $$96 \times 0.08 = 7.68$$ 11. **Step 4:** Add tax to discounted price: $$96 + 7.68 = 103.68$$ 12. **Final answer:** The final price after discount and tax is $103.68$. This approach helps solve inequalities and apply consumer arithmetic in real-life budgeting and shopping scenarios.