|3x - 2| = -5 How many distinct real solutions does the given equation have?...

1. TRANSLATE the problem information

• Given equation: \(|3\mathrm{x} - 2| = -5\)
• Need to find: How many distinct real solutions exist

2. INFER what absolute value means

• The expression \(|3\mathrm{x} - 2|\) represents the distance from \((3\mathrm{x} - 2)\) to zero on the number line
• Distance is always non-negative, so \(|3\mathrm{x} - 2| \geq 0\) for any real number x

3. APPLY CONSTRAINTS to determine if solution exists

• We need \(|3\mathrm{x} - 2| = -5\)
• But \(|3\mathrm{x} - 2| \geq 0\) (from step 2)
• Since \(-5 \lt 0\), there's no way a non-negative quantity can equal -5

4. INFER the final conclusion

• The equation \(|3\mathrm{x} - 2| = -5\) has no solutions
• It's impossible for any real number x to satisfy this equation

Answer: D (Zero)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that the equation is impossible and instead trying to solve using standard absolute value case analysis.

Students think: "I'll solve this like any absolute value equation by considering two cases: \(3\mathrm{x} - 2 = -5\) and \(3\mathrm{x} - 2 = 5\)." They solve both cases and get \(\mathrm{x} = -1\) and \(\mathrm{x} = \frac{7}{3}\), then conclude there are exactly two solutions.

This may lead them to select Choice B (Exactly two).

Second Most Common Error:

Missing conceptual knowledge: Not understanding that absolute value represents distance and is therefore always non-negative.

Students may recognize something is "off" about the negative right side but can't articulate why this makes the equation impossible. They might guess randomly among the choices or assume there's some advanced technique they don't know.

This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students truly understand what absolute value means geometrically (distance) rather than just mechanically knowing how to solve absolute value equations through case analysis.