Solve for x: 2x + 7 = 3x - 4

The variable x equals 11 in the equation 2x + 7 = 3x - 4 when algebraic manipulation isolates the variable by subtracting 2x from both sides to consolidate variable terms, then adding 4 to both sides to eliminate the constant on the right, yielding x = 11 as the solution that satisfies equality upon verification.

A. 3

Substituting x = 3 produces 2(3) + 7 = 6 + 7 = 13 on the left versus 3(3) - 4 = 9 - 4 = 5 on the right revealing a substantial 8-unit disparity that violates equation equality. This error likely originates from incorrectly combining constants across the equality sign without maintaining balanced operations: adding 7 + 4 = 11 then dividing by an erroneous coefficient sum (2 + 3 = 5) to yield 11 ÷ 5 ≈ 2.2 rounded to 3. Alternatively, students might execute 3x - 2x = x correctly but then miscalculate 7 - 4 = 3 instead of 7 + 4 = 11, demonstrating sign error during constant transposition a pervasive misconception where learners treat all transposed terms as subtraction regardless of original sign. This distractor effectively identifies students with partial equation-solving knowledge who recognize variable consolidation but lack precision in sign management during term transposition, a deficiency requiring explicit attention to the principle that transposing terms across equality requires sign inversion to maintain balanced equivalence.

B. -3

This value yields 2(-3) + 7 = -6 + 7 = 1 versus 3(-3) - 4 = -9 - 4 = -13 upon substitution producing values with opposite signs and 14-unit separation. The negative result suggests systematic sign confusion during solution execution, possibly from incorrectly moving the +7 to the right as -7 while simultaneously moving -4 to the left as +4 but then executing 4 - 7 = -3 instead of 7 + 4 = 11. Another plausible pathway involves solving 2x - 3x = -4 - 7 → -x = -11 then erroneously concluding x = -11 ÷ 1 = -11 but mis-transcribing as -3 through digit substitution. This option proves particularly instructive for diagnosing sign management deficiencies in algebraic manipulation especially critical when negative coefficients or constants appear, as sign errors produce solutions with incorrect polarity that completely reverse solution meaning in applied contexts like temperature differentials, financial debits/credits, or coordinate geometry positioning.

C. 11

This value correctly satisfies the equation through systematic algebraic isolation executed via two complementary pathways with verification. Primary solution sequence: subtract 2x from both sides (2x + 7 - 2x = 3x - 4 - 2x → 7 = x - 4), then add 4 to both sides (7 + 4 = x - 4 + 4 → 11 = x). Alternative sequence: subtract 3x from both sides (2x + 7 - 3x = 3x - 4 - 3x → -x + 7 = -4), subtract 7 ( -x = -11), multiply by -1 (x = 11). Verification substitution: left side 2(11) + 7 = 22 + 7 = 29; right side 3(11) - 4 = 33 - 4 = 29; equality confirmed. This solution demonstrates comprehensive mastery of equation-solving principles: maintaining equality through balanced operations, consolidating like terms across the equality sign, managing sign changes during transposition, and verifying solutions to detect procedural errors. The integer solution reflects appropriate problem design where coefficients yield whole-number results, building student confidence while reinforcing core algebraic manipulation without fractional complexity that might obscure procedural understanding.

D. -11

Substituting x = -11 yields 2(-11) + 7 = -22 + 7 = -15 versus 3(-11) - 4 = -33 - 4 = -37 revealing 22-unit disparity despite both sides being negative. This error represents a classic sign inversion mistake where students correctly execute -x = -11 during solution but then erroneously conclude x = -11 instead of multiplying both sides by -1 to obtain x = 11. The error pathway likely involves solving 2x - 3x = -4 - 7 → -x = -11 then stopping prematurely without executing the final sign correction step. This "stopped-short" pattern reveals incomplete understanding of variable isolation requirements when negative coefficients appear a critical deficiency as negative coefficients frequently emerge in real-world modeling (depreciation rates, temperature decreases, debt accumulation) where sign errors completely reverse solution interpretation. The proximity of -11 to the intermediate calculation step makes this an exceptionally effective distractor targeting students with partial procedural knowledge who recognize equation manipulation steps but lack comprehensive solution sequencing awareness through final isolation completion.

Conclusion

The solution x = 11 emerges through rigorous application of algebraic manipulation principles maintaining equality through balanced operations while systematically isolating the variable. This problem exemplifies foundational linear equation solving essential for advanced algebraic reasoning, function analysis, and mathematical modeling where unknown quantities must be determined through structured transformation sequences. Mastery demands understanding that equation equality requires identical operations on both sides, recognizing efficient term consolidation strategies (moving smaller coefficient to larger), managing sign changes during transposition with precision, and verifying solutions to detect undetected procedural errors. The sign inversion error represented by option D proves particularly instructive, highlighting how partial procedural knowledge produces intermediate values with inverted polarity mistaken for final solutions a misconception requiring explicit instructional attention to develop complete solution pathway awareness including final sign correction when negative coefficients remain after consolidation. Students developing automaticity with these structured approaches build robust algebraic reasoning capabilities transferable across mathematical domains where equation solving forms the backbone of quantitative problem resolution in physics, economics, engineering, and data science contexts.