Recognize an Identity after Distribution

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: How many real solutions does 2(x+2)=2x+4 have? Candidate answers: (A) Every real x (B) Exactly one, x=2 (C) Exactly two, x=0 and x=2 (D) No real x.
• Independent check inputs: How many real solutions does 3(x+2)=3x+6 have? Candidate answers: (A) Every real x (B) Exactly one, x=2 (C) Exactly two, x=0 and x=2 (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. Equivalent left and right expressions produce a true equality for every allowed input. If all variable terms cancel and equal constants remain, the equation has infinitely many real solutions.

Here is the method on the supplied teaching case. Distribute the left side: 2x+4. Both sides are identical, so every real value satisfies the equation. No division by a zero coefficient is needed.

Cancellation does not automatically mean no solution. Inspect the constant statement that remains; a true equality means every allowed value, while a false equality means none.

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 9(x+2)=9x+18 have? Candidate answers: (A) Every real x (B) Exactly one, x=2 (C) Exactly two, x=0 and x=2 (D) No real x. Show the mathematical reason, not only the selected answer. Expected result: Every real x. Distribute the left side: 9x+18. Both sides are identical, so every real value satisfies the equation. No division by a zero coefficient is needed.

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