Combine a Cost Cap with Whole-Number Counts

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A fictional model has a fixed cost $7 plus $5 per item x. Spending must be at most $17. Which whole-number counts are allowed? Candidate answers: (A) 0 through 1 (B) 0 through 2 (C) 0 through 3 (D) 2 or more.
• Independent check inputs: A fictional model has a fixed cost $7 plus $5 per item x. Spending must be at most $22. Which whole-number counts are allowed? Candidate answers: (A) 0 through 2 (B) 0 through 3 (C) 0 through 4 (D) 3 or more.

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. An at-most constraint includes its boundary and can be expressed with ≤. A solution must satisfy the algebraic inequality and the context’s additional nonnegative integer restriction.

Here is the method on the supplied teaching case. Write 5x+7≤17. Subtract 7 and divide by 5 to obtain x≤2. A count must also be a nonnegative integer.

A fraction may solve the algebraic inequality but cannot represent a count of whole items. Include zero if the fictional context permits buying nothing; do not silently round an upper bound upward.

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: A fictional model has a fixed cost $7 plus $5 per item x. Spending must be at most $52. Which whole-number counts are allowed? Candidate answers: (A) 0 through 10 (B) 0 through 8 (C) 0 through 9 (D) 9 or more. Show the mathematical reason, not only the selected answer. Expected result: 0 through 9. Write 5x+7≤52. Subtract 7 and divide by 5 to obtain x≤9. A count must also be a nonnegative integer.

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