Equation Case Files Worksheet
CASE FILE #8-MS Algebraic Forensics Bureau
EQUATION CASE FILES
Investigating Multi-Step Equations & Solution Classifications
Agent:
Date:
Period:
OUTCOME DECODER
1. Unique Solution Isolates to \(x = c\). Exactly one valid value satisfies the case.
2. No Solution (\(\emptyset\)) Variables cancel to a contradiction (e.g., \(0 = 7\)). False trail!
3. Infinite Solutions (\(\mathbb{R}\)) Variables cancel to an identity (e.g., \(4 = 4\)). True for all real numbers.
PART 1
Scaffolded Field Inquiries
Distribute, combine like terms, and balance both sides
CASE #01 Guided Solve
\(3(2x - 4) = 4x + 10\)
Clue: Distribute the 3, collect variables on the left, then isolate \(x\).
One Solution: \(x =\) _________
No Solution (\(\emptyset\)) Infinitely Many Solutions
CASE #02 Guided Solve
\(-5x + 9 = 2x - 12\)
Clue: Watch negative signs carefully when transferring variable terms.
One Solution: \(x =\) _________
No Solution (\(\emptyset\)) Infinitely Many Solutions
CASE #03 Special Case Check
\(4(x + 3) = 2(2x + 6)\)
Clue: Distribute both sides completely before collecting like terms.
One Solution: \(x =\) _________
No Solution (\(\emptyset\)) Infinitely Many Solutions
CASE #04 Special Case Check
\(6x - 8 = 2(3x - 1)\)
Clue: Distribute the 2, then subtract \(6x\) from both sides. What remains?
One Solution: \(x =\) _________
No Solution (\(\emptyset\)) Infinitely Many Solutions
Equation Case Files • Page 1 of 2 Verify all steps with inverse operations
CASE FILE #8-MS Part 2 & 3: Field Investigation & Error Forensics
Agent Docket Cont.
PART 2
Mixed Field Cases (Extended Work Area)
Simplify each side, show complete algebraic work, and classify
CASE #05 Mixed Multi-Step
\(2(4x - 3) - 5x = 3x - 6\)
Investigation Note: Distribute first, then combine like terms on the left side before moving variables.
One Solution: \(x =\) _________
No Solution (\(\emptyset\)) Infinitely Many Solutions
CASE #06 Mixed Multi-Step
\(8 - 3(2x - 5) = -6x + 20\)
Investigation Note: Be extremely careful distributing \(-3\) to both terms inside parentheses.
One Solution: \(x =\) _________
No Solution (\(\emptyset\)) Infinitely Many Solutions
PART 3
Forensic Error Analysis
Review suspect case files, identify the algebraic error, and provide the valid resolution
INCIDENT LOG A Distribution Breach
Flawed Suspect File:
Equation: \(5(x - 3) = 2x + 6\)
Step 1: \(5x - 3 = 2x + 6\)
Step 2: \(3x - 3 = 6\)
Step 3: \(3x = 9 \rightarrow x = 3\)
1. Identify the specific error in Step 1:
2. Provide the correct algebraic solution:
INCIDENT LOG B Classification Breach
Flawed Suspect File:
Equation: \(2(3x + 4) = 6x + 8\)
Step 1: \(6x + 8 = 6x + 8\)
Step 2: \(8 = 8\)
Suspect Claim: "Answer is \(x = 0\) because variables disappeared."
1. Explain why concluding "\(x = 0\)" is mathematically incorrect:
2. State the valid solution classification and what it means:
Equation Case Files • Page 2 of 2 Case Solved: Classification Complete
Equation Case Files Answer Key
TEACHER DOSSIER Master Key & Scoring Guide
EQUATION CASE FILES • ANSWER KEY
Complete Step-by-Step Solutions & Pedagogical Diagnostics
Clearance Status VERIFIED SOLUTIONS 8th / 9th Grade Algebra
GRADING PROTOCOL
Check Line 1: Ensure distributive property was applied to all terms, preserving signs.
Check Balancing: Inverse operations applied equally to both sides of the equals sign.
Classification: Student must state specific outcome, not just write an intermediate line.
PART 1 KEY
Scaffolded Field Inquiries: Solutions
Cases 01–04 Fully Worked
CASE #01 ONE SOLUTION
\(3(2x - 4) = 4x + 10\)
Step 1: \(6x - 12 = 4x + 10\)
Step 2: \(2x - 12 = 10\) (-4x both sides)
Step 3: \(2x = 22\) (+12 both sides)
Final: \(x = 11\)
Common Pitfall: Forgetting to multiply \(3 \times (-4)\), mistakenly writing \(6x - 4 = 4x + 10\).
CASE #02 ONE SOLUTION
\(-5x + 9 = 2x - 12\)
Step 1: \(9 = 7x - 12\) (+5x both sides)
Step 2: \(21 = 7x\) (+12 both sides)
Step 3: \(3 = x\) (\(\div 7\) both sides)
Final: \(x = 3\)
Teaching Strategy: Encourage moving the smaller variable term (\(-5x\)) to keep the coefficient of \(x\) positive.
CASE #03 INFINITE SOLUTIONS
\(4(x + 3) = 2(2x + 6)\)
Step 1: \(4x + 12 = 4x + 12\) (Distribute)
Step 2: \(12 = 12\) (-4x both sides)
Result: Identity (Statement always true)
Outcome: Infinitely Many Solutions (\(\mathbb{R}\))
Concept: Any value chosen for \(x\) satisfies the equation because both expressions are fundamentally identical.
CASE #04 NO SOLUTION (\(\emptyset\))
\(6x - 8 = 2(3x - 1)\)
Step 1: \(6x - 8 = 6x - 2\) (Distribute 2)
Step 2: \(-8 = -2\) (-6x both sides)
Result: Contradiction (\(-8 \neq -2\) is false)
Outcome: No Solution (\(\emptyset\))
Common Pitfall: Students writing "x = 0" when variables eliminate. Emphasize that a false constant equation means no solution.
Equation Case Files Teacher Key • Page 1 of 2 Part 1 Review Complete
TEACHER DOSSIER Part 2 & 3: Mixed & Forensic Analysis Key
Page 2 Master Key
PART 2 KEY
Mixed Field Cases (Expanded Solutions)
Cases 05–06 Fully Worked
CASE #05 INFINITE SOLUTIONS
\(2(4x - 3) - 5x = 3x - 6\)
1. Distribute: \(8x - 6 - 5x = 3x - 6\)
