Question:Consider the equation |2x - 7| = -5.How many distinct real solutions does this equation have?012Infinitely many

1. TRANSLATE the problem information

• Given equation: \(|2x - 7| = -5\)
• Find: Number of distinct real solutions

2. INFER the fundamental constraint

• Key insight: Absolute value represents distance from zero
• Distance is always non-negative: \(|\mathrm{anything}| \geq 0\)
• This means \(|2x - 7| \geq 0\) for ALL real values of x

3. APPLY CONSTRAINTS to determine solution existence

• We need \(|2x - 7| = -5\)
• But \(|2x - 7| \geq 0\) and \(-5 \lt 0\)
• Since a non-negative number cannot equal a negative number, this equation is impossible

Answer: A (0 solutions)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to solve by cases without recognizing the fundamental impossibility.

They might write: "Case 1: \(2x - 7 = -5\), so \(x = 1\)" and "Case 2: \(2x - 7 = 5\), so \(x = 6\)" then conclude there are 2 solutions. This ignores that absolute value equations of the form \(|\mathrm{expression}| = \mathrm{negative\ number}\) have no solutions by definition.

This may lead them to select Choice C (2 solutions).

Second Most Common Error:

Missing conceptual knowledge: Not understanding that absolute values are always non-negative.

Students might treat \(|2x - 7|\) like a regular algebraic expression and attempt various algebraic manipulations without recognizing the constraint that absolute values cannot be negative.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students truly understand what absolute value means (distance/non-negative) rather than just memorizing solution procedures. The trap is rushing into case-by-case solving without first checking if the equation is even possible.