Choose an Efficient Balanced Operation

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Solve 2(x−1)=4. Candidate answers: (A) 2 (B) 3 (C) 4 (D) 6.
• Independent check inputs: Solve 3(x−1)=9. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8.

Learn the idea

Solving means finding inputs that make a relationship true. Balanced operations preserve equality, whereas inequality direction must reverse when multiplication or division by a negative number reverses number-line order. A nonzero factor multiplying an entire group can be divided out before expanding. This often shortens a solution while keeping the same balance and respecting the group’s structure.

Here is the method on the supplied teaching case. Divide both sides by 2: x−1=2. Add 1 to both sides: x=3. Substitution gives 2(3−1)=4.

Division must affect the entire side of the equation. Once the outside factor is removed, undo the inner addition or subtraction; substitute the solution before treating it as established.

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: Solve 9(x−1)=81. Candidate answers: (A) 10 (B) 11 (C) 20 (D) 9. Show the mathematical reason, not only the selected answer. Expected result: 10. Divide both sides by 9: x−1=9. Add 1 to both sides: x=10. Substitution gives 9(10−1)=81.

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