Question:|5 - 4y| = 15What is the value of the positive solution to the equation above?

1. TRANSLATE the problem information

• Given equation: \(|5 - 4\mathrm{y}| = 15\)
• Find: The positive solution

2. CONSIDER ALL CASES for the absolute value

• Since \(|5 - 4\mathrm{y}| = 15\), this means the expression \((5 - 4\mathrm{y})\) is either 15 units or -15 units from zero
• This gives us two possible cases:
  • Case 1: \(5 - 4\mathrm{y} = 15\)
  • Case 2: \(5 - 4\mathrm{y} = -15\)

3. SIMPLIFY Case 1: \(5 - 4\mathrm{y} = 15\)

• Subtract 5 from both sides: \(-4\mathrm{y} = 10\)
• Divide by -4: \(\mathrm{y} = -2.5\)

4. SIMPLIFY Case 2: \(5 - 4\mathrm{y} = -15\)

• Subtract 5 from both sides: \(-4\mathrm{y} = -20\)
• Divide by -4: \(\mathrm{y} = 5\)

5. APPLY CONSTRAINTS to select the final answer

• We found two solutions: \(\mathrm{y} = -2.5\) and \(\mathrm{y} = 5\)
• The question asks for the positive solution
• Since \(5 \gt 0\), the answer is 5

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often solve only one case of the absolute value equation, typically just setting \(5 - 4\mathrm{y} = 15\) and stopping there.

They might think: "The absolute value equals 15, so what's inside must equal 15." This gives them \(\mathrm{y} = -2.5\) as their only answer. Since -2.5 is negative, they might think there's no positive solution or get confused about what the question is asking.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly find both solutions (\(\mathrm{y} = -2.5\) and \(\mathrm{y} = 5\)) but then give both answers instead of identifying which one is positive.

They might write "\(\mathrm{y} = -2.5\) or \(\mathrm{y} = 5\)" without recognizing that the question specifically asks for the positive solution only.

The Bottom Line:

Absolute value equations always require checking two scenarios, and students must pay careful attention to any constraints on which solution(s) to report. The key insight is that \(|\mathrm{expression}| = \mathrm{positive\ number}\) always means \(\mathrm{expression} = \pm\mathrm{positive\ number}\).