Let's analyze the questions under "Solve each equation. Check your answer" and solve them step-by-step.

1. $4 \sqrt[3]{2x + 11} = 12$

Step 1: Isolate the radical term. [ \frac{4 \sqrt[3]{2x + 11}}{4} = \frac{12}{4} \implies \sqrt[3]{2x + 11} = 3 ]

Step 2: Eliminate the cube root by raising both sides to the power of 3. $\left( \sqrt[3]{2x + 11} \right)^3 = 3^3$ $2x + 11 = 27$

Step 3: Solve for $x$. $2x = 27 - 11 \implies 2x = 16$ $x = \frac{16}{2} = 8$

Step 4: Check the solution. Substitute $x = 8$ back into the original equation: [ 4 \sqrt[3]{2(8) + 11} = 4 \sqrt[3]{16 + 11} = 4 \sqrt[3]{27} = 4(3) = 12 ] The solution is correct: $x = 8$.

2. $5 + \sqrt{x - 3} = 9$

Step 1: Isolate the square root. $\sqrt{x - 3} = 9 - 5 \implies \sqrt{x - 3} = 4$

Step 2: Square both sides to eliminate the square root. $\left( \sqrt{x - 3} \right)^2 = 4^2$ $x - 3 = 16$

Step 3: Solve for $x$. $x = 16 + 3 \implies x = 19$

Step 4: Check the solution. Substitute $x = 19$ back into the original equation: $5 + \sqrt{19 - 3} = 5 + \sqrt{16} = 5 + 4 = 9$ The solution is correct: $x = 19$.

3. $2 \sqrt{x + 4} = 10$

Step 1: Isolate the square root. $\frac{2 \sqrt{x + 4}}{2} = \frac{10}{2} \implies \sqrt{x + 4} = 5$

Step 2: Square both sides to eliminate the square root. $\left( \sqrt{x + 4} \right)^2 = 5^2$ $x + 4 = 25$

Step 3: Solve for $x$. $x = 25 - 4 \implies x = 21$

Step 4: Check the solution. Substitute $x = 21$ back into the original equation: $2 \sqrt{21 + 4} = 2 \sqrt{25} = 2(5) = 10$ The solution is correct: $x = 21$.

Final Solutions:

1. $x = 8$
2. $x = 19$
3. $x = 21$

Let me know if you need further clarification on any step!

Here are 5 related questions for practice:

1. Solve $3 \sqrt{x - 2} = 15$.
2. Solve $7 + \sqrt{2x - 5} = 10$.
3. Solve $2 \sqrt[3]{x + 1} = 6$.
4. Solve $5 \sqrt{x + 4} = 20$.
5. Solve $\sqrt[3]{3x - 7} = 2$.

Tip: Always isolate the radical first and then eliminate it using powers to simplify.