Rational Expressions, Equations, and Variation: cumulative review

Compare the conclusions of all three worked cases with this lesson outcome—Assemble an LCD using every factor at its greatest required power. Explain what remains invariant across them.

Write a complete attempt before opening the response guide.

Exit check: solve and verify without referring to the displayed steps. Find the least common denominator of $6x^{2}y$ and $15xy^{3}$.

Write a complete attempt before opening the response guide.

Exit check: solve and verify without referring to the displayed steps. Find the LCD of $x^{2} - 9$ and $x^{2} + 6x + 9$.

Write a complete attempt before opening the response guide.

Classify the mathematical object and requested action in this lesson case: Simplify $\frac{2}{x - 1} - \frac{3}{x + 2}$.

Write a complete attempt before opening the response guide.

State the central definition behind this outcome: Create equivalent rational expressions with a common denominator before combining numerators.

Write a complete attempt before opening the response guide.

Before calculating, list every sign, endpoint, unit, grouping, or domain condition that can affect: Simplify $\frac{2}{x - 1} - \frac{3}{x + 2}$.

Write a complete attempt before opening the response guide.

Explain why this opening move is valid: Use $(x - 1)(x + 2)$ as the common denominator and state restrictions.

Write a complete attempt before opening the response guide.

Simplify$$\frac{2}{x - 1} - \frac{3}{x + 2}$$

Write a complete attempt before opening the response guide.

Simplify$$\frac{3}{x - 2} + \frac{5}{x + 1}$$

Write a complete attempt before opening the response guide.

Simplify$$\frac{2x}{x^{2} - 4} - \frac{1}{x + 2}$$

Write a complete attempt before opening the response guide.

Verify the proposed result “$\frac{7 - x}{(x - 1)(x + 2)},$ with $x \ne 1, -2$.” against the original statement.

Write a complete attempt before opening the response guide.

Complete the calculation after “Use LCD $(x - 2)(x + 1)$.” in this problem: Simplify $\frac{3}{x - 2} + \frac{5}{x + 1}$.

Write a complete attempt before opening the response guide.

Name and justify the most efficient first move, then solve: Simplify $\frac{2x}{x^{2} - 4} - \frac{1}{x + 2}$.

Write a complete attempt before opening the response guide.

Compare the methods used in these two cases and identify the structural reason they differ: Simplify $\frac{3}{x - 2} + \frac{5}{x + 1}$. Simplify $\frac{2x}{x^{2} - 4} - \frac{1}{x + 2}$.

Write a complete attempt before opening the response guide.

Create the representation most useful for checking this result: Simplify $\frac{3}{x - 2} + \frac{5}{x + 1}$. Show the original restriction set, factored form, simplified form, and graph features such as holes or asymptotes when relevant.

Write a complete attempt before opening the response guide.

A learner reports “$\frac{7 - x}{(x - 1)(x + 2)},$ with $x \ne 1, -2$.” but omits the original-condition check. Explain the risk before deciding whether the result is supported.

Write a complete attempt before opening the response guide.

Repair a solution that skips “Rewrite the second fraction with numerator $x - 2$ over the common denominator.” while solving: Simplify $\frac{2x}{x^{2} - 4} - \frac{1}{x + 2}$.

Write a complete attempt before opening the response guide.

In this addition and subtraction case, change one numerical value, solve the revised problem, and identify which parts of the original method still apply: Simplify $\frac{2}{x - 1} - \frac{3}{x + 2}$.

Write a complete attempt before opening the response guide.

Connect the opening situation “Combine two fractional rates or parts.” to the algebraic structure used in the worked case. Define quantities and units before writing any equation.

Write a complete attempt before opening the response guide.

Explain why the method for addition and subtraction is valid here and name one nearby problem where it would not apply.

Write a complete attempt before opening the response guide.

Compare the conclusions of all three worked cases with this lesson outcome—Create equivalent rational expressions with a common denominator before combining numerators. Explain what remains invariant across them.

Write a complete attempt before opening the response guide.

Exit check: solve and verify without referring to the displayed steps. Simplify $\frac{3}{x - 2} + \frac{5}{x + 1}$.

Write a complete attempt before opening the response guide.

Exit check: solve and verify without referring to the displayed steps. Simplify $\frac{2x}{x^{2} - 4} - \frac{1}{x + 2}$.

Write a complete attempt before opening the response guide.

Classify the mathematical object and requested action in this lesson case: Simplify $\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$.

Write a complete attempt before opening the response guide.

State the central definition behind this outcome: Simplify a fraction containing fractions by multiplying numerator and denominator by a common LCD.

Write a complete attempt before opening the response guide.

Before calculating, list every sign, endpoint, unit, grouping, or domain condition that can affect: Simplify $\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$.

Write a complete attempt before opening the response guide.

Explain why this opening move is valid: State $x \ne 0, y \ne 0,$ and require the complex-expression denominator to be nonzero.

Write a complete attempt before opening the response guide.

Simplify$$\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$$

Write a complete attempt before opening the response guide.

Simplify$$\frac{\frac{2}{x} + \frac{3}{y}}{\frac{1}{x} - \frac{1}{y}}$$

Write a complete attempt before opening the response guide.

Simplify$$\frac{\frac{2}{x + 1}}{\frac{3}{x - 2} - 1}$$

Write a complete attempt before opening the response guide.

Verify the proposed result “$\frac{x + y}{(y -}$ x), with $x \ne 0, y \ne 0,$ and $x \ne y$.” against the original statement.

Write a complete attempt before opening the response guide.

Complete the calculation after “Use xy as the inner common denominator.” in this problem: Simplify $\frac{\frac{2}{x} + \frac{3}{y}}{\frac{1}{x} - \frac{1}{y}}$.

Write a complete attempt before opening the response guide.

Name and justify the most efficient first move, then solve: Simplify $\frac{\frac{2}{x + 1}}{\frac{3}{x - 2} - 1}$.

Write a complete attempt before opening the response guide.

Compare the methods used in these two cases and identify the structural reason they differ: Simplify $\frac{\frac{2}{x} + \frac{3}{y}}{\frac{1}{x} - \frac{1}{y}}$. Simplify $\frac{\frac{2}{x + 1}}{\frac{3}{x - 2} - 1}$.

Write a complete attempt before opening the response guide.

Create the representation most useful for checking this result: Simplify $\frac{\frac{2}{x} + \frac{3}{y}}{\frac{1}{x} - \frac{1}{y}}$. Show the original restriction set, factored form, simplified form, and graph features such as holes or asymptotes when relevant.

Write a complete attempt before opening the response guide.