Recognize a Contradiction after Cancellation

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: How many real solutions does 2x+3=2x+4 have? Candidate answers: (A) Every real x (B) Exactly one, x=1 (C) Exactly one, x=3 (D) No real x.
• Independent check inputs: How many real solutions does 3x+4=3x+5 have? Candidate answers: (A) Every real x (B) Exactly one, x=1 (C) Exactly one, x=4 (D) No real x.

Learn the idea

An equation’s solution count depends on its entire structure. Balanced simplification may isolate a variable, reduce to an identity, or expose a contradiction; each outcome has a different interpretation. Equal variable coefficients can cancel even when the equation is false for every input. Unequal remaining constants produce a contradiction that no variable value can repair.

Here is the method on the supplied teaching case. Subtract 2x from both sides. This leaves 3=4, which is false. A contradiction means the original equation has no solution.

Do not divide by a zero coefficient to invent a numerical answer. Write the reduced constant comparison explicitly, and distinguish a false statement from a statement that is true for every input.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: How many real solutions does 9x+10=9x+11 have? Candidate answers: (A) Every real x (B) Exactly one, x=1 (C) Exactly one, x=10 (D) No real x. Show the mathematical reason, not only the selected answer. Expected result: No real x. Subtract 9x from both sides. This leaves 10=11, which is false. A contradiction means the original equation has no solution.

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