Solve: |2x - 5| = 7

The equation |2x - 5| = 7 has two solutions, x = -1 and x = 6, because absolute value equations yield two cases: the expression inside equals the positive value (2x - 5 = 7) or the negative value (2x - 5 = -7), each solved separately to produce x = 6 and x = -1 respectively, both satisfying the original equation upon verification.

A. x = 1 or x = 6

Substituting x = 1 yields |2(1) - 5| = |2 - 5| = |-3| = 3 ≠ 7, failing verification, while x = 6 correctly yields |12 - 5| = |7| = 7. This partial correctness reveals a systematic error in solving the negative case: students likely set up 2x - 5 = -7 correctly but then solved 2x = -2 to get x = -1 yet mis-transcribed as x = 1 during answer selection possibly through sign omission during transcription or cognitive interference from the positive solution's magnitude. Another plausible pathway involves solving 2x - 5 = 7 → x = 6 correctly but then for the negative case executing 2x + 5 = 7 (incorrect sign change) → 2x = 2 → x = 1. This distractor effectively identifies learners with partial case analysis knowledge who recognize two solutions are required but introduce sign errors during negative case setup or solution a transitional deficiency requiring explicit emphasis on the precise negative case formulation (expression = -constant, not expression with sign-flipped terms = constant) and verification protocols to catch transcription errors before final answer submission.

B. x = -1 or x = 6

This solution set correctly satisfies the absolute value equation through systematic case analysis with comprehensive verification. Positive case: 2x - 5 = 7 → 2x = 12 → x = 6; verification |2(6) - 5| = |12 - 5| = |7| = 7 ✓. Negative case: 2x - 5 = -7 → 2x = -2 → x = -1; verification |2(-1) - 5| = |-2 - 5| = |-7| = 7 ✓. Graphical interpretation: the expression 2x - 5 represents a line with slope 2; absolute value creates V-shape vertex at x = 2.5; horizontal line y = 7 intersects V-shape at two points corresponding to x = -1 and x = 6. Distance interpretation: |2x - 5| = 7 means 2x is 7 units from 5 on the number line → 2x = 5 + 7 = 12 or 2x = 5 - 7 = -2 → x = 6 or x = -1. This solution demonstrates comprehensive mastery of absolute value equation solving including case decomposition protocol, sign management during negative case setup, algebraic isolation maintaining equality balance, and dual verification ensuring both solutions satisfy the original equation a critical step as extraneous solutions sometimes emerge in more complex absolute value scenarios. These integrated competencies prove essential for inequality solving, piecewise function analysis, distance problems in coordinate geometry, and error margin calculations where absolute value models tolerance ranges requiring dual-boundary determination.

C. x = 1 or x = -6

Both values fail verification: x = 1 yields |2 - 5| = 3 ≠ 7; x = -6 yields |-12 - 5| = |-17| = 17 ≠ 7. This complete failure suggests fundamental misunderstanding of absolute value case setup possibly solving |2x| - 5 = 7 (misplaced absolute value) → |2x| = 12 → x = ±6, then adjusting signs arbitrarily to reach ±1 and ∓6. Another plausible error pathway involves solving 2x - 5 = 7 → x = 6 correctly but then for negative case executing -(2x) - 5 = 7 → -2x = 12 → x = -6, then arbitrarily changing 6 to 1 through digit substitution. The symmetric magnitude pattern (1 and 6) with incorrect signs reveals possible random sign assignment without verification a dangerous approach where learners recognize two solutions are needed but lack systematic case methodology, resorting to guesswork when procedural knowledge is incomplete. This distractor effectively identifies students with severe conceptual gaps in absolute value interpretation, treating it as optional sign variation rather than distance-from-zero representation requiring structured case analysis a deficiency requiring explicit reteaching of absolute value as distance metric with concrete number line demonstrations before advancing to equation solving.

D. x = -1 or x = -6

While x = -1 correctly satisfies the equation, x = -6 fails verification: |2(-6) - 5| = |-12 - 5| = 17 ≠ 7. This partial correctness reveals error in positive case solution: students likely solved the negative case correctly (2x - 5 = -7 → x = -1) but then for positive case executed 2x - 5 = -7 again (repeating negative case) or solved 2x + 5 = 7 → 2x = 2 → x = 1 then mis-transcribed as x = -6 through unstructured adjustment. Another plausible pathway involves solving 2x = 12 → x = 6 correctly but then applying negative sign universally to both solutions (x = -6, x = -1) through overgeneralization of sign patterns. The consistent negative polarity across both solutions suggests possible misconception that absolute value equations always yield negative solutions or confusion with inequality directionality where negative multipliers flip inequality signs a conceptual conflation requiring explicit differentiation between equation solving (yielding specific values) versus inequality solving (yielding intervals with directionality considerations).

Conclusion

The solution set x = -1 or x = 6 emerges through rigorous application of absolute value case decomposition recognizing that |expression| = constant requires solving both expression = constant and expression = -constant, followed by algebraic isolation and mandatory verification of both solutions in the original equation. This problem reinforces critical algebraic reasoning competencies essential across mathematical domains: understanding absolute value as distance representation requiring dual-case analysis, executing precise sign management during case setup (particularly the negative case where the entire expression equals negative constant, not individual term sign flips), maintaining solution integrity through verification protocols that catch transcription errors or extraneous solutions, and interpreting solutions geometrically as intersection points between absolute value graphs and horizontal lines. Mastery of these integrated skills proves indispensable for calculus (limit definitions involving epsilon-delta with absolute values), physics (error analysis with tolerance ranges), optimization (constraint satisfaction with deviation bounds), and computer science (boundary condition testing) where absolute value models deviation magnitudes requiring symmetric boundary determination. The distractors strategically target pervasive error patterns including sign transcription errors, case setup misconceptions, and verification omission highlighting the necessity of explicit case labeling ("Case 1: positive", "Case 2: negative") and mandatory substitution verification before final answer submission to prevent undetected errors that compromise solution validity in increasingly complex absolute value scenarios including inequalities and nested absolute values.