|2k - 5| = 9 What is one possible value for k? 7 -2 2 14...

1. TRANSLATE the problem information

• Given: \(|2\mathrm{k} - 5| = 9\)
• Find: One possible value for k
• What this tells us: We have an absolute value equation to solve

2. CONSIDER ALL CASES for the absolute value

• Key insight: When \(|\mathrm{expression}| = \mathrm{positive\ number}\), the expression inside can equal that positive number OR its negative
• This gives us two cases to solve:
  • Case 1: \(2\mathrm{k} - 5 = 9\) (when the expression is positive or zero)
  • Case 2: \(2\mathrm{k} - 5 = -9\) (when the expression is negative)

3. SIMPLIFY Case 1: \(2\mathrm{k} - 5 = 9\)

• Add 5 to both sides: \(2\mathrm{k} = 14\)
• Divide by 2: \(\mathrm{k} = 7\)

4. SIMPLIFY Case 2: \(2\mathrm{k} - 5 = -9\)

• Add 5 to both sides: \(2\mathrm{k} = -4\)
• Divide by 2: \(\mathrm{k} = -2\)

5. TRANSLATE back to answer format

• Both \(\mathrm{k} = 7\) and \(\mathrm{k} = -2\) are valid solutions
• Since the question asks for "one possible value," either is mathematically correct
• Choice (A) is \(\mathrm{k} = 7\), Choice (B) is \(\mathrm{k} = -2\)

Answer: B (-2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students only solve one case, typically the "positive" case where \(2\mathrm{k} - 5 = 9\), getting \(\mathrm{k} = 7\). They forget that absolute value equations generally have two solutions because \(|\mathrm{x}| = \mathrm{a}\) means \(\mathrm{x}\) could be \(\mathrm{a}\) or \(-\mathrm{a}\).

This leads them to confidently select Choice A (7) without realizing they've missed half the solution.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what an absolute value represents or confuse the setup. They might incorrectly think \(|2\mathrm{k} - 5| = 9\) means \(2\mathrm{k} - 5 = \pm9\) should be solved as one equation, leading to algebraic confusion.

This causes them to get stuck and guess randomly among the choices.

The Bottom Line:

The key challenge is recognizing that absolute value equations split into two separate linear equations. Students who treat this as just another linear equation miss the fundamental nature of absolute values and only find one of the two correct solutions.