The equation \(4(\mathrm{x} + 2) = 4\mathrm{x} + 8\) is given. How many real solutions does the equation have?Exactly oneNo...

1. SIMPLIFY the left side of the equation

• Given: \(4(\mathrm{x} + 2) = 4\mathrm{x} + 8\)
• Apply distributive property to expand \(4(\mathrm{x} + 2)\):
  \(4(\mathrm{x} + 2) = 4\mathrm{x} + 8\)
• Now our equation looks like: \(4\mathrm{x} + 8 = 4\mathrm{x} + 8\)

2. SIMPLIFY by eliminating like terms

• Subtract 4x from both sides:
  \(4\mathrm{x} + 8 - 4\mathrm{x} = 4\mathrm{x} + 8 - 4\mathrm{x}\)
• This gives us: \(8 = 8\)

3. INFER what this result means

• When we get a statement like \(8 = 8\), this is called an identity
• An identity is always true, no matter what value we substitute for x
• This means every real number satisfies the original equation
• Therefore, there are infinitely many solutions

Answer: C (Infinitely many)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly simplify to get \(8 = 8\), but then misinterpret what this means. They think "8 = 8 doesn't tell me what x equals, so there's no solution."

This confusion about the meaning of identities leads them to select Choice B (No solution) instead of recognizing that an identity means infinitely many solutions work.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make an algebraic error during the expansion or when subtracting terms. For example, they might incorrectly expand \(4(\mathrm{x} + 2)\) as \(4\mathrm{x} + 2\), leading to the equation \(4\mathrm{x} + 2 = 4\mathrm{x} + 8\), which would give \(-6 = 0\) (no solution).

This algebraic mistake causes them to select Choice B (No solution) for the wrong reasons.

The Bottom Line:

The key challenge is recognizing that when algebraic manipulation leads to a true statement (like \(8 = 8\)), this indicates that the original equation is an identity with infinitely many solutions, not a contradiction with no solutions.