1. **Problem a:** Solve $3(y + 1)^2 - 5 = 43$. 2. **Rewrite the equation:** Add 5 to both sides: $$3(y + 1)^2 = 48$$ 3. **Undo the multiplication:** Divide both sides by 3: $$ (y + 1)^2 = 16 $$ 4. **Undo the square:** Take the square root of both sides: $$ y + 1 = \pm 4 $$ 5. **Solve for $y$:** - If $y + 1 = 4$, then $y = 3$ - If $y + 1 = -4$, then $y = -5$ --- 1. **Problem b:** Solve $\sqrt{1 - 4x} = 10$. 2. **Undo the square root:** Square both sides: $$ 1 - 4x = 100 $$ 3. **Solve for $x$:** $$ -4x = 99 $$ $$ x = -\frac{99}{4} = -24.75 $$ --- 1. **Problem c:** Solve $\frac{6y - 1}{y} - 3 = 2$. 2. **Rewrite the equation:** Add 3 to both sides: $$ \frac{6y - 1}{y} = 5 $$ 3. **Multiply both sides by $y$ to clear the denominator:** $$ 6y - 1 = 5y $$ 4. **Solve for $y$:** $$ 6y - 5y = 1 $$ $$ y = 1 $$ --- 1. **Problem d:** Solve $\sqrt[3]{1 - 2x} = 3$. 2. **Undo the cube root:** Cube both sides: $$ 1 - 2x = 27 $$ 3. **Solve for $x$:** $$ -2x = 26 $$ $$ x = -13 $$ --- **Final answers:** - a) $y = 3$ or $y = -5$ - b) $x = -24.75$ - c) $y = 1$ - d) $x = -13$