Rational Expressions, Equations, and Variation: mastery answer key

A learner reports “$x \ne 2$ and $x \ne -5$.” but omits the original-condition check. Explain the risk before deciding whether the result is supported.

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Repair a solution that skips “Substitute $x = 3$ into numerator and denominator.” while solving: Find the restriction and evaluate $\frac{2x - 5}{x + 4}$ at $x = 3$.

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In this rational expressions and restrictions case, change one numerical value, solve the revised problem, and identify which parts of the original method still apply: State the domain restrictions of $\frac{x + 4}{(x - 2)(x + 5)}$.

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Connect the opening situation “Identify forbidden settings in a rate or formula.” to the algebraic structure used in the worked case. Define quantities and units before writing any equation.

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Explain why the method for rational expressions and restrictions is valid here and name one nearby problem where it would not apply.

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Compare the conclusions of all three worked cases with this lesson outcome—Determine where a denominator is zero and state the valid input set. Explain what remains invariant across them.

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Exit check: solve and verify without referring to the displayed steps. State the domain restrictions of $\frac{x + 1}{x^{2} - 5x + 6}$.

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Exit check: solve and verify without referring to the displayed steps. Find the restriction and evaluate $\frac{2x - 5}{x + 4}$ at $x = 3$.

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Classify the mathematical object and requested action in this lesson case: Simplify $\frac{x^{2} - 9}{x^{2} - x - 6}$ and preserve all original restrictions.

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State the central definition behind this outcome: Factor and cancel common factors without cancelling terms or erasing restrictions.

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Before calculating, list every sign, endpoint, unit, grouping, or domain condition that can affect: Simplify $\frac{x^{2} - 9}{x^{2} - x - 6}$ and preserve all original restrictions.

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Explain why this opening move is valid: Factor the numerator as $(x - 3)(x + 3)$.

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Simplify $\frac{x^{2} - 9}{x^{2} - x - 6}$ and preserve all original restrictions.

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Simplify $\frac{4x^{2} - 25}{2x^{2} + 9x + 10}$ and preserve restrictions.

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Explain and correct the false cancellation $\frac{x + 6}{x} = 6$.

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Verify the proposed result “$\frac{x + 3}{x + 2},$ with $x \ne 3$ and $x \ne -2$.” against the original statement.

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Complete the calculation after “Factor the numerator as $(2x - 5)(2x + 5)$.” in this problem: Simplify $\frac{4x^{2} - 25}{2x^{2} + 9x + 10}$ and preserve restrictions.

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Name and justify the most efficient first move, then solve: Explain and correct the false cancellation $\frac{x + 6}{x} = 6$.

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Compare the methods used in these two cases and identify the structural reason they differ: Simplify $\frac{4x^{2} - 25}{2x^{2} + 9x + 10}$ and preserve restrictions. Explain and correct the false cancellation $\frac{x + 6}{x} = 6$.

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Create the representation most useful for checking this result: Simplify $\frac{4x^{2} - 25}{2x^{2} + 9x + 10}$ and preserve restrictions. Show the original restriction set, factored form, simplified form, and graph features such as holes or asymptotes when relevant.

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A learner reports “$\frac{x + 3}{x + 2},$ with $x \ne 3$ and $x \ne -2$.” but omits the original-condition check. Explain the risk before deciding whether the result is supported.

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Repair a solution that skips “Rewrite $\frac{x + 6}{x}$ as $\frac{x}{x} + \frac{6}{x}$ for $x \ne 0$.” while solving: Explain and correct the false cancellation $\frac{x + 6}{x} = 6$.

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In this simplifying rational expressions case, change one numerical value, solve the revised problem, and identify which parts of the original method still apply: Simplify $\frac{x^{2} - 9}{x^{2} - x - 6}$ and preserve all original restrictions.

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Connect the opening situation “Simplify a formula with a shared factor.” to the algebraic structure used in the worked case. Define quantities and units before writing any equation.

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