Solve Equations C380E3

1. **Problem:** Solve for $h$ in the formula $$S = 2\pi r(r + h)$$ 2. **Formula:** The surface area $S$ of a cylinder is given by $$S = 2\pi r(r + h)$$ where $r$ is the radius and $h$ is the height. 3. **Step 1:** Start with the equation: $$S = 2\pi r(r + h)$$ 4. **Step 2:** Divide both sides by $2\pi r$ to isolate $(r + h)$: $$\frac{S}{2\pi r} = r + h$$ 5. **Step 3:** Subtract $r$ from both sides to solve for $h$: $$h = \frac{S}{2\pi r} - r$$ --- 1. **Problem:** Solve for $x$ in the inequality $$4x + 11 \leq 2(x - 3) + 1$$ 2. **Step 1:** Expand the right side: $$4x + 11 \leq 2x - 6 + 1$$ $$4x + 11 \leq 2x - 5$$ 3. **Step 2:** Subtract $2x$ from both sides: $$4x - 2x + 11 \leq -5$$ $$2x + 11 \leq -5$$ 4. **Step 3:** Subtract $11$ from both sides: $$2x \leq -5 - 11$$ $$2x \leq -16$$ 5. **Step 4:** Divide both sides by $\cancel{2}$ to get: $$x \leq \frac{\cancel{-16}}{\cancel{2}}$$ $$x \leq -8$$ --- 1. **Problem:** Solve for $x$ in the equation $$3x - 4 = 17$$ 2. **Step 1:** Add $4$ to both sides: $$3x = 17 + 4$$ $$3x = 21$$ 3. **Step 2:** Divide both sides by $\cancel{3}$: $$x = \frac{\cancel{21}}{\cancel{3}}$$ $$x = 7$$ --- 1. **Problem:** Solve for $x$ in the inequality $$|5x - 2| > 13$$ 2. **Step 1:** Recall that $|A| > B$ means $A > B$ or $A < -B$. 3. **Step 2:** Set up two inequalities: $$5x - 2 > 13 \quad \text{or} \quad 5x - 2 < -13$$ 4. **Step 3:** Solve the first inequality: $$5x > 13 + 2$$ $$5x > 15$$ $$x > \frac{15}{5}$$ $$x > 3$$ 5. **Step 4:** Solve the second inequality: $$5x < -13 + 2$$ $$5x < -11$$ $$x < \frac{-11}{5}$$ 6. **Solution:** $$x < -\frac{11}{5} \quad \text{or} \quad x > 3$$ --- 1. **Problem:** Solve for $y$ in the equation $$5y + 11 = 9y - 13$$ 2. **Step 1:** Subtract $5y$ from both sides: $$11 = 4y - 13$$ 3. **Step 2:** Add $13$ to both sides: $$11 + 13 = 4y$$ $$24 = 4y$$ 4. **Step 3:** Divide both sides by $\cancel{4}$: $$y = \frac{\cancel{24}}{\cancel{4}}$$ $$y = 6$$