5|x| = 45 What is the positive solution to the given equation?...

1. SIMPLIFY the equation to isolate the absolute value

• Given: \(5|\mathrm{x}| = 45\)
• Divide both sides by 5: \(|\mathrm{x}| = 9\)

2. INFER what the absolute value equation tells us

• When \(|\mathrm{x}| = 9\), we need to think about what values of x have an absolute value of 9
• By definition of absolute value: if \(|\mathrm{x}| = 9\), then \(\mathrm{x} = 9\) or \(\mathrm{x} = -9\)
• This gives us two potential solutions

3. APPLY CONSTRAINTS to select the correct answer

• The problem specifically asks for "the positive solution"
• From our two solutions (9 and -9), only 9 is positive
• Therefore, \(\mathrm{x} = 9\)

Answer: 9

Alternative acceptable forms: Just 9 (since it's already simplified)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about absolute value definition: Students may think \(|\mathrm{x}| = 9\) means \(\mathrm{x} = 9\) only, forgetting that absolute value equations typically have two solutions.

This incomplete understanding leads them to find the correct answer by luck, but they miss the underlying mathematical reasoning about why absolute value equations have two solutions.

Second Most Common Error:

Weak APPLY CONSTRAINTS reasoning: Students correctly find both solutions (\(\mathrm{x} = 9\) and \(\mathrm{x} = -9\)) but then provide both answers instead of selecting only the positive one as requested.

This may lead them to write "9 and -9" or select multiple answers if this were a multiple choice question, rather than recognizing the problem asks specifically for the positive solution.

The Bottom Line:

This problem tests whether students truly understand that absolute value creates a "distance from zero" relationship, which naturally leads to two solutions, and whether they can follow specific instructions to select only one of those solutions.