Compound Inequalities 73Bc3E

1. **State the problem:** Solve the compound inequality: $$12x + 4 < 5x + 25 \quad \text{AND} \quad 4 + 6x \leq 15x - 5$$ 2. **Solve the first inequality:** $$12x + 4 < 5x + 25$$ Subtract $5x$ from both sides: $$12x - 5x + 4 < 25$$ $$7x + 4 < 25$$ Subtract $4$ from both sides: $$7x + \cancel{4} - \cancel{4} < 25 - 4$$ $$7x < 21$$ Divide both sides by $7$: $$\frac{\cancel{7}x}{\cancel{7}} < \frac{21}{7}$$ $$x < 3$$ 3. **Solve the second inequality:** $$4 + 6x \leq 15x - 5$$ Subtract $6x$ from both sides: $$4 + \cancel{6x} - \cancel{6x} \leq 15x - 6x - 5$$ $$4 \leq 9x - 5$$ Add $5$ to both sides: $$4 + 5 \leq 9x - 5 + 5$$ $$9 \leq 9x$$ Divide both sides by $9$: $$\frac{9}{9} \leq \frac{9x}{9}$$ $$1 \leq x$$ 4. **Combine the two inequalities:** $$x < 3 \quad \text{AND} \quad x \geq 1$$ 5. **Write the solution in interval notation:** $$[1, 3)$$ **Final answer:** The solution to the compound inequality is $$[1, 3)$$.