|x + 5| + |y - 8| = 9 The graph of the given equation in the xy-plane is a...

1. TRANSLATE the equation form

• Given: \(|\mathrm{x} + 5| + |\mathrm{y} - 8| = 9\)
• Rewrite in standard form: \(|\mathrm{x} - (-5)| + |\mathrm{y} - 8| = 9\)
• This tells us: Diamond centered at (-5, 8) with total "radius" of 9

2. INFER the strategy for finding possible x-values

• To find the range of possible x-coordinates, we need the leftmost and rightmost points of the diamond
• These extreme points occur when all 9 units of distance are used purely horizontally

3. SIMPLIFY to find the x-range

• Leftmost point: \(\mathrm{x} = \mathrm{center\_x} - 9 = -5 - 9 = -14\)
• Rightmost point: \(\mathrm{x} = \mathrm{center\_x} + 9 = -5 + 9 = 4\)
• Therefore: \(-14 \leq \mathrm{x} \leq 4\)

4. APPLY CONSTRAINTS to eliminate answer choices

• Check each choice against our range [-14, 4]:
  • A. -15: Too far left \(\lt -14\) ✗
  • B. -13: Within range ✓
  • C. 5: Too far right \(\gt 4\) ✗
  • D. 8: Too far right \(\gt 4\) ✗

5. Verify our answer works

• If \(\mathrm{a} = -13\): \(|-13 + 5| = 8\), so \(|\mathrm{y} - 8| = 9 - 8 = 1\)
• This gives \(\mathrm{y} = 7\) or \(\mathrm{y} = 9\), both valid solutions

Answer: B. -13

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE execution: Students misread \(|\mathrm{x} + 5|\) as \(|\mathrm{x} - 5|\) and identify the center as (5, 8) instead of (-5, 8).

With wrong center (5, 8), they calculate x-range as [5-9, 5+9] = [-4, 14]. Looking at choices, both A (-15) and B (-13) appear outside this wrong range, while C (5) and D (8) fall within it. This creates confusion about which choices are valid, leading to guessing or selecting Choice C (5).

Second Most Common Error:

Weak INFER reasoning: Students recognize the diamond shape but don't understand how to find the x-coordinate range systematically.

They might try plugging answer choices directly into the equation without establishing the valid range first. This leads to computational confusion and typically results in selecting the first choice that seems to "work" when substituted, potentially picking Choice D (8) or giving up and guessing.

The Bottom Line:

The key insight is that absolute value equations create geometric constraints - you must first understand the shape and its boundaries before checking individual points.