Week 11 — Cancel, Combine, Check: A Short Rational-Expression Report — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

Assignment 10 of the term — simplify, combine, and solve four short rational problems, explain what each answer means, and tell someone new to algebra why two pumps working together finish sooner than either alone.

This week you learned to find the excluded values of a rational expression, to simplify it by cancelling common factors and never terms, to multiply, divide, add, and subtract rational expressions, to solve rational equations and reject extraneous solutions, and to model work-rate settings. In Week 6 you factored polynomials such as x² − 6x + 5 = (x − 1)(x − 5), and in Week 2 you cleared fractions with a least common denominator. In this assignment you work four short problems, interpret each one, and explain to someone who has never studied algebra why two pumps together finish sooner than either alone. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

The task

You will use four problems from this week: the week's quotient (x² − 6x + 5) ⁄ (x − 1), the difference x ⁄ (x² − 4) − 1 ⁄ (x − 2), the equation 1 ⁄ (x − 1) + 1 = 2 ⁄ (x² − 1), and a pump problem. In the pump problem, one pump alone empties a flooded basement in 12 hours and a second pump alone empties it in 24 hours, and both run at steady rates.

Work these four problems, showing your arithmetic:

1. Simplify the week's quotient (x² − 6x + 5) ⁄ (x − 1). Its graph has a hole at x = 1. What is the y-coordinate of the hole?
2. Subtract x ⁄ (x² − 4) − 1 ⁄ (x − 2) and write the difference as one simplified fraction. What does it equal at x = 6?
3. Solve 1 ⁄ (x − 1) + 1 = 2 ⁄ (x² − 1). Which candidate survives the check?
4. Two pumps empty a flooded basement. One alone takes 12 hours and the other alone takes 24 hours. How many hours do they take working together?

Beside each answer, add one brief interpretation: what that value shows and one use or limit of it. For problem 1, show the factoring and the cancel, write the restriction, and say why x = 1 gives a hole and not a zero. For problem 2, show the minus sign reaching every term of the second numerator, and check your value against the original expression at x = 6. For problem 3, name the excluded values first, say which candidate is extraneous and why, and check the one that survives. For problem 4, write each pump's rate, add the rates, and check that the two pumps' shares make one whole job.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

Explain it to a non-expert

A housemate is helping you deal with the flooded basement and has never studied algebra. They think the two pumps together will take 36 hours, or maybe 18, the average. Write a 3–5 sentence message for your housemate: why adding or averaging the times cannot be right, how thinking about each pump's share per hour gives the time together, and one honest limit of the model, such as what it assumes about the pumps. Explain any technical term in everyday words. More than one good limit can be defended.

For a different, ungraded example, imagine two friends washing a big stack of dishes: one alone needs 30 minutes and the other alone needs 60 minutes. Model answer: "Working together can only be faster than the faster friend alone, so the answer has to be under 30 minutes. In one minute the first friend does one thirtieth of the stack and the second does one sixtieth, and together that is one twentieth of the stack each minute. So the two of them finish in 20 minutes. That assumes they never get in each other's way at one sink."

Use that pattern for the pumps, with your own explanation.

Deliverables

Submit a single rational-expression report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of what that value shows and one use or limit of it;
• your message for your housemate, explaining why the times do not add, how the shares per hour give the time together, and one honest limit; and
• a one-line restatement of the expressions and equation you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled sketch with a short note. Every form is graded by the same scoring criteria. Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a free graphing calculator such as Desmos to check your arithmetic and your answers. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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