2|4 - x| + 3|4 - x| = 25 What is the positive solution to the given equation?...

1. SIMPLIFY the equation by combining like terms

• Given: \(2|4 - \mathrm{x}| + 3|4 - \mathrm{x}| = 25\)
• Since both terms contain the same absolute value expression \(|4 - \mathrm{x}|\), we can combine:
  • \(2|4 - \mathrm{x}| + 3|4 - \mathrm{x}| = (2 + 3)|4 - \mathrm{x}| = 5|4 - \mathrm{x}|\)
• This gives us: \(5|4 - \mathrm{x}| = 25\)

2. SIMPLIFY further by isolating the absolute value

• Divide both sides by 5:
  • \(|4 - \mathrm{x}| = 5\)

3. CONSIDER ALL CASES for the absolute value equation

• When \(|4 - \mathrm{x}| = 5\), we have two possibilities:
  • Case 1: \(4 - \mathrm{x} = 5\) (when the expression inside is positive)
  • Case 2: \(4 - \mathrm{x} = -5\) (when the expression inside is negative)

4. SIMPLIFY each case to find x-values

• Case 1: \(4 - \mathrm{x} = 5\)
  • Subtract 4: \(-\mathrm{x} = 1\)
  • Multiply by -1: \(\mathrm{x} = -1\)

• Case 2: \(4 - \mathrm{x} = -5\)
  • Subtract 4: \(-\mathrm{x} = -9\)
  • Multiply by -1: \(\mathrm{x} = 9\)

5. APPLY CONSTRAINTS to select the final answer

• We found two solutions: \(\mathrm{x} = -1\) and \(\mathrm{x} = 9\)
• The problem asks for "the positive solution"
• Since \(9 \gt 0\) and \(-1 \lt 0\), the positive solution is \(\mathrm{x} = 9\)

Answer: 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students might not recognize that \(2|4-\mathrm{x}| + 3|4-\mathrm{x}|\) can be combined into \(5|4-\mathrm{x}|\), instead trying to solve each absolute value term separately or getting confused about how to handle multiple absolute value expressions.

This leads to much more complicated work and often abandoning the systematic solution, resulting in confusion and guessing.

Second Most Common Error:

Missing CONSIDER ALL CASES reasoning: Students solve \(|4-\mathrm{x}| = 5\) but only consider one case (usually \(4-\mathrm{x} = 5\)), missing the second case entirely.

This may lead them to find only \(\mathrm{x} = -1\), and since that's negative, they might think there's no positive solution or guess randomly among answer choices.

The Bottom Line:

This problem tests whether students can efficiently combine like terms with absolute values and systematically work through both cases that absolute value equations create. The key insight is recognizing that multiple absolute value terms with the same expression can be combined just like any other like terms.