If sqrt(3x + 7) = 4, which of the following is a possible value of x? 1 3 5 9...

1. TRANSLATE the problem information

• Given: \(\sqrt{3x + 7} = 4\)
• Find: The value of x from the answer choices

2. INFER the solution strategy

• To solve an equation with a square root, we need to eliminate the radical
• The most direct approach: square both sides of the equation
• This will give us a linear equation we can solve normally

3. SIMPLIFY by squaring both sides

• Square the left side: \((\sqrt{3x + 7})^2 = 3x + 7\)
• Square the right side: \(4^2 = 16\)
• Result: \(3x + 7 = 16\)

4. SIMPLIFY the linear equation

• Subtract 7 from both sides: \(3x = 16 - 7 = 9\)
• Divide both sides by 3: \(x = 3\)

5. Verify the solution

• Check: \(\sqrt{3(3) + 7} = \sqrt{9 + 7} = \sqrt{16} = 4\) ✓
• Our solution works!

Answer: B) 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when solving the linear equation \(3x = 9\).

Some students might incorrectly calculate \(16 - 7 = 8\) instead of 9, leading to \(3x = 8\) and \(x = 8/3\). Since \(8/3 \approx 2.67\), this doesn't match any answer choice exactly, leading to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize they need to square both sides to eliminate the square root.

Instead, they might try to work with the radical directly or attempt other approaches that don't lead to a clear path forward. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students understand the fundamental strategy for solving radical equations: eliminate the radical first, then solve the resulting simpler equation. The algebraic steps themselves are straightforward once the strategy is clear.