1. **Problem 1:** Solve the inequality $3x + 5 \leq 3x - 3$. 2. **Step 1:** Write the inequality: $$3x + 5 \leq 3x - 3$$ 3. **Step 2:** Subtract $3x$ from both sides: $$3x - 3x + 5 \leq 3x - 3x - 3$$ $$5 \leq -3$$ 4. **Step 3:** This is a false statement since $5$ is not less than or equal to $-3$. 5. **Conclusion:** There is no solution to this inequality. --- 1. **Problem 2:** Solve the inequality $x + 2 \geq 4x - 7$. 2. **Step 1:** Write the inequality: $$x + 2 \geq 4x - 7$$ 3. **Step 2:** Subtract $4x$ from both sides: $$x - 4x + 2 \geq 4x - 4x - 7$$ $$-3x + 2 \geq -7$$ 4. **Step 3:** Subtract $2$ from both sides: $$-3x \geq -9$$ 5. **Step 4:** Divide both sides by $-3$ and reverse the inequality sign: $$x \leq 3$$ 6. **Step 5:** Solution set is $x \leq 3$. --- 1. **Problem 3:** Solve the inequality $6y - 30 > 20 + y$. 2. **Step 1:** Write the inequality: $$6y - 30 > 20 + y$$ 3. **Step 2:** Subtract $y$ from both sides: $$6y - y - 30 > 20$$ $$5y - 30 > 20$$ 4. **Step 3:** Add $30$ to both sides: $$5y > 50$$ 5. **Step 4:** Divide both sides by $5$: $$y > 10$$ 6. **Step 5:** Solution set is $y > 10$. --- 1. **Problem 4:** Solve the inequality $5y - 20 < 10y$. 2. **Step 1:** Write the inequality: $$5y - 20 < 10y$$ 3. **Step 2:** Subtract $5y$ from both sides: $$-20 < 5y$$ 4. **Step 3:** Divide both sides by $5$: $$-4 < y$$ 5. **Step 4:** Rewrite as: $$y > -4$$ 6. **Step 5:** Solution set is $y > -4$. --- 1. **Problem 5:** Solve the inequality $\frac{4x}{2} + 3 \geq 6x - 5$. 2. **Step 1:** Simplify the left side: $$2x + 3 \geq 6x - 5$$ 3. **Step 2:** Subtract $6x$ from both sides: $$2x - 6x + 3 \geq -5$$ $$-4x + 3 \geq -5$$ 4. **Step 3:** Subtract $3$ from both sides: $$-4x \geq -8$$ 5. **Step 4:** Divide both sides by $-4$ and reverse the inequality sign: $$x \leq 2$$ 6. **Step 5:** Solution set is $x \leq 2$. Each problem was solved step-by-step with explanations and the solution sets are: 1) No solution 2) $x \leq 3$ 3) $y > 10$ 4) $y > -4$ 5) $x \leq 2$