For real number r, consider the equation |r| = 2|r|. How many real solutions does this equation have?

1. TRANSLATE the problem information

• Given equation: \(|\mathrm{r}| = 2|\mathrm{r}|\)
• Need to find: How many real solutions exist
• This is asking us to solve for all possible values of r

2. SIMPLIFY by isolating the absolute value term

• Subtract \(2|\mathrm{r}|\) from both sides:
  \(|\mathrm{r}| - 2|\mathrm{r}| = 2|\mathrm{r}| - 2|\mathrm{r}|\)
  \(-|\mathrm{r}| = 0\)
  \(|\mathrm{r}| = 0\)

3. INFER what \(|\mathrm{r}| = 0\) means for r

• Since absolute value represents distance from zero
• \(|\mathrm{r}| = 0\) means r is exactly 0 units away from zero
• Therefore: \(\mathrm{r} = 0\)

4. Verify the solution

• Check: \(|0| = 2|0|\) → \(0 = 2(0)\) → \(0 = 0\) ✓
• The equation holds true

Answer: B (1 solution)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students attempt to divide both sides by \(|\mathrm{r}|\) to get \(1 = 2\)

Students think: "If \(|\mathrm{r}| = 2|\mathrm{r}|\), I'll divide both sides by \(|\mathrm{r}|\) to get \(1 = 2\), which is impossible, so there are no solutions."

This reasoning fails because division by \(|\mathrm{r}|\) is only valid when \(|\mathrm{r}| \neq 0\). Since \(\mathrm{r} = 0\) is actually the solution, dividing by \(|\mathrm{r}|\) eliminates the very solution we're looking for! This leads them to select Choice A (0 solutions).

The Bottom Line:

The key insight is recognizing that when an equation involves absolute values, you must consider the case where the absolute value equals zero. Attempting to divide by the absolute value expression can eliminate valid solutions through division by zero.