Algebra Flashcards: Operating With Rational Expressions

What this deck covers

This deck focuses on Operating With Rational Expressions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is $\frac{x}{x+2}\cdot\frac{x+2}{3}$ simplified?

Answer: $\frac{x}{3}$ (with $x\ne -2$). The $(x+2)$ terms cancel, leaving $\frac{x}{3}$.

Flashcard 2: What is $\frac{4}{x^2-9}+\frac{1}{x+3}$ written as a single fraction?

Answer: $\frac{x+7}{x^2-9}$. Use LCD $x^2-9=(x+3)(x-3)$ to combine the fractions.

Flashcard 3: What restriction must be stated for a rational expression $\frac{p(x)}{q(x)}$?

Answer: $q(x)\ne 0$ (exclude values that make the denominator $0$). The denominator cannot equal zero to avoid undefined expressions.

Flashcard 4: What is $\frac{1}{x+1}-\frac{1}{x-1}$ written as a single fraction?

Answer: $\frac{-2}{x^2-1}$. Use LCD $x^2-1$ to get $\frac{x-1-(x+1)}{x^2-1}$.

Flashcard 5: What is the quotient $\frac{2}{x}\div\frac{3}{x+1}$ simplified?

Answer: $\frac{2(x+1)}{3x}$. Multiply by reciprocal: $\frac{2}{x}\cdot\frac{x+1}{3}$.

Flashcard 6: What is $\frac{3}{x+2}+\frac{1}{x+2}$ simplified?

Answer: $\frac{4}{x+2}$. Add numerators since denominators are identical.

Flashcard 7: What is $\frac{2}{x}-\frac{1}{2x}$ written as a single fraction?

Answer: $\frac{3}{2x}$. Use LCD $2x$: $\frac{4}{2x}-\frac{1}{2x}=\frac{3}{2x}$.

Flashcard 8: What is $\frac{x-3}{x^2-9}$ simplified?

Answer: $\frac{1}{x+3}$ (with $x\ne 3$ and $x\ne -3$). Factor $x^2-9=(x-3)(x+1)$ and cancel $(x-3)$.

Flashcard 9: What is $\frac{1}{x+1}+\frac{1}{x-1}$ written as a single fraction?

Answer: $\frac{2x}{x^2-1}$. Use LCD $(x+1)(x-1)=x^2-1$ to get $\frac{x-1+x+1}{x^2-1}$

Flashcard 10: What is $\frac{2}{x-3}-\frac{x}{x-3}$ simplified?

Answer: $\frac{2-x}{x-3}$. Subtract numerators: $\frac{2-x}{x-3}$.

Flashcard 11: What is $\frac{x-3}{x^2-9}$ simplified?

Answer: $\frac{1}{x+3}$ (with $x\ne 3$ and $x\ne -3$). Factor $x^2-9=(x-3)(x+1)$ and cancel $(x-3)$.

Flashcard 12: What does it mean for rational expressions to be closed under multiplication?

Answer: The product of two rational expressions is a rational expression. Multiplying rational expressions produces another rational expression.

Flashcard 13: What is $\frac{x}{x+1}+\frac{1}{x+1}$ simplified?

Answer: $\frac{x+1}{x+1}=1$ (with $x\ne -1$). Adding the fractions gives $\frac{x+1}{x+1}$ which simplifies to $1$.

Flashcard 14: What is $\frac{5}{x}-\frac{2}{x}$ simplified?

Answer: $\frac{3}{x}$. Subtract numerators when denominators are the same.

Flashcard 15: What is the least common denominator (LCD) of $\frac{1}{x}$ and $\frac{1}{x-2}$?

Answer: $x(x-2)$. Take the product of distinct denominators for the LCD.

Flashcard 16: What is $\frac{(x-1)(x+4)}{x-1}$ simplified?

Answer: $x+4$ (with $x \neq 1$). Cancel the common factor $(x-1)$.

Flashcard 17: What is a rational expression in terms of polynomials $p(x)$ and $q(x)$?

Answer: $\frac{p(x)}{q(x)}$ where $p(x),q(x)$ are polynomials and $q(x)\ne 0$. This is the standard form where polynomials create a fraction.

Flashcard 18: What is $\frac{2x}{x^2}$ simplified?

Answer: $\frac{2}{x}$ (with $x\ne 0$). Cancel one factor of $x$ from numerator and denominator.

Flashcard 19: What is $\frac{1}{x-3}-\frac{1}{x+3}$ written as a single fraction?

Answer: $\frac{6}{x^2-9}$. Use LCD $x^2-9$ to get $\frac{x+3-(x-3)}{x^2-9}=\frac{6}{x^2-9}$.

Flashcard 20: What is the rule for dividing rational expressions $\frac{a}{b}\div\frac{c}{d}$?

Answer: $\frac{a}{b}\cdot\frac{d}{c}$ with $b\ne 0$ and $c\ne 0$ and $d\ne 0$. Division becomes multiplication by the reciprocal of the divisor.

Flashcard 21: What is $\frac{x}{x-2}+\frac{2}{x-2}$ simplified?

Answer: $\frac{x+2}{x-2}$. Add numerators since denominators are the same.

Flashcard 22: What is $\frac{1}{x}+\frac{2}{3x}$ written as a single fraction?

Answer: $\frac{5}{3x}$. Use LCD $3x$: $\frac{3}{3x}+\frac{2}{3x}=\frac{5}{3x}$.

Flashcard 23: What is $\frac{a}{b}\cdot\frac{c}{d}$ simplified in one step?

Answer: $\frac{ac}{bd}$ (with $b\ne 0$ and $d\ne 0$). Multiply numerators together and denominators together.

Flashcard 24: What is $\frac{2}{x}+\frac{3}{x}$ simplified?

Answer: $\frac{5}{x}$. Add numerators when denominators are the same.

Flashcard 25: What condition is required for closure under division of rational expressions?

Answer: Division is allowed only by a nonzero rational expression. Cannot divide by zero, so the divisor must be nonzero.

Flashcard 26: What is $\frac{1}{x}+\frac{1}{x-1}$ written as a single fraction?

Answer: $\frac{2x-1}{x(x-1)}$. Use LCD $x(x-1)$ and combine: $\frac{x-1+x}{x(x-1)}$.

Flashcard 27: Identify the missing condition for $\frac{a}{b}\div\frac{c}{d}$ to be defined.

Answer: $\frac{c}{d}\ne 0$ (equivalently $c\ne 0$ and $d\ne 0$). The divisor must be nonzero for division to be defined.

Flashcard 28: What does it mean for rational expressions to be closed under addition and subtraction?

Answer: The sum or difference of two rational expressions is a rational expression. Adding or subtracting rational expressions yields another rational expression.

Flashcard 29: What is $\frac{x^2-1}{x+1}$ simplified?

Answer: $x-1$ (with $x\ne -1$). Factor $x^2-1=(x-1)(x+1)$ and cancel $(x+1)$.

Flashcard 30: What is $\frac{1}{x}\div\frac{1}{x-3}$ simplified?

Answer: $\frac{x-3}{x}$ (with $x\ne 0$ and $x\ne 3$). Multiply by reciprocal: $\frac{1}{x}\cdot\frac{x-3}{1}$.

Flashcard 31: What is $\frac{x^2-1}{x+1}\div(x-1)$ simplified?

Answer: $1$ (with $x\neq -1$ and $x\neq 1$). Rewrite division as multiplication by reciprocal, then simplify.

Flashcard 32: What is $\frac{4}{x^2-9}+\frac{1}{x+3}$ written as a single fraction?

Answer: $\frac{x+7}{x^2-9}$. Use LCD $x^2-9=(x+3)(x-3)$ to combine the fractions.

Flashcard 33: What is $\frac{x}{x+2}\cdot\frac{x+2}{3}$ simplified?

Answer: $\frac{x}{3}$ (with $x\ne -2$). The $(x+2)$ terms cancel, leaving $\frac{x}{3}$.

Flashcard 34: What is the domain restriction for $\frac{1}{x-3}$?

Answer: $x\ne 3$. Setting the denominator equal to zero: $x-3=0$ gives $x=3$.

Flashcard 35: What is $\frac{x^2+5x}{x}$ simplified?

Answer: $x+5$ (with $x\ne 0$). Factor out $x$ from numerator: $\frac{x(x+5)}{x}$.

Flashcard 36: What is $\frac{1}{x+1}+\frac{1}{x-1}$ written as a single fraction?

Answer: $\frac{2x}{x^2-1}$. Use LCD $(x+1)(x-1)=x^2-1$ to get $\frac{x-1+x+1}{x^2-1}$.

Flashcard 37: Identify the missing condition for $\frac{a}{b}\div\frac{c}{d}$ to be defined.

Answer: $\frac{c}{d}\ne 0$ (equivalently $c\ne 0$ and $d\ne 0$). The divisor must be nonzero for division to be defined.

Flashcard 38: What is $\frac{x+2}{x}-\frac{2}{x}$ simplified?

Answer: $\frac{x}{x}=1$ (with $x\ne 0$). Subtracting gives $\frac{x+2-2}{x}=\frac{x}{x}=1$.

Flashcard 39: What is $\frac{2}{x}-\frac{1}{2x}$ written as a single fraction?

Answer: $\frac{3}{2x}$. Use LCD $2x$: $\frac{4}{2x}-\frac{1}{2x}=\frac{3}{2x}$.

Flashcard 40: What is the product $\frac{2}{x}\cdot\frac{3}{x+1}$ simplified?

Answer: $\frac{6}{x(x+1)}$. Multiply numerators and denominators directly.

Flashcard 41: What is the LCD of $\frac{1}{x^2}$ and $\frac{1}{x}$?

Answer: $x^2$. Use the highest power of $x$ that appears in either denominator.

Flashcard 42: What is the reciprocal of the rational expression $\frac{a}{b}$ (with $a\ne 0$ and $b\ne 0$)?

Answer: $\frac{b}{a}$. Flip the numerator and denominator to get the reciprocal.

Flashcard 43: What does it mean for rational expressions to be closed under multiplication?

Answer: The product of two rational expressions is a rational expression. Multiplying rational expressions produces another rational expression.

Flashcard 44: What is $\frac{x^2-4}{x-2}$ simplified?

Answer: $x+2$ (with $x \neq 2$). Factor $x^2-4=(x-2)(x+2)$ and cancel $(x-2)$.

Flashcard 45: What is $\frac{x}{x^2-1}+\frac{1}{x-1}$ written as a single fraction?

Answer: $\frac{2x+1}{x^2-1}$. Rewrite $\frac{1}{x-1}$ with LCD $x^2-1$ then add.

Flashcard 46: What is $\frac{x^2-9}{x-3}$ simplified?

Answer: $x+3$ (with $x\ne 3$). Factor $x^2-9=(x-3)(x+1)$ and cancel $(x-3)$.

Flashcard 47: What is $\frac{a}{b}+\frac{c}{b}$ simplified, assuming $b\ne 0$?

Answer: $\frac{a+c}{b}$. Combine fractions with the same denominator.

Flashcard 48: What is $\frac{x^2-9}{x}\cdot\frac{x}{x-3}$ simplified?

Answer: $x+3$ (with $x\ne 0$ and $x\ne 3$). Factor $x^2-9=(x-3)(x+1)$, then cancel common factors.

Flashcard 49: What is $\frac{x+2}{x}-\frac{2}{x}$ simplified?

Answer: $\frac{x}{x}=1$ (with $x\ne 0$). Subtracting gives $\frac{x+2-2}{x}=\frac{x}{x}=1$.

Flashcard 50: What is $\frac{1}{x}+\frac{2}{3x}$ written as a single fraction?

Answer: $\frac{5}{3x}$. Use LCD $3x$: $\frac{3}{3x}+\frac{2}{3x}=\frac{5}{3x}$.

Flashcard 51: What is $\frac{a}{b}-\frac{c}{b}$ simplified, assuming $b\ne 0$?

Answer: $\frac{a-c}{b}$. Combine fractions with the same denominator.

Flashcard 52: What is $\frac{3x^2y}{6xy^2}$ simplified?

Answer: $\frac{x}{2y}$ (with $x\ne 0$ and $y\ne 0$). Cancel common factors: $3x$ and $y$.

Flashcard 53: What is $\frac{a}{b}\cdot\frac{c}{d}$ simplified in one step?

Answer: $\frac{ac}{bd}$ (with $b\ne 0$ and $d\ne 0$). Multiply numerators together and denominators together.

Flashcard 54: What is $\frac{1}{x}\div\frac{1}{x-3}$ simplified?

Answer: $\frac{x-3}{x}$ (with $x\ne 0$ and $x\ne 3$). Multiply by reciprocal: $\frac{1}{x}\cdot\frac{x-3}{1}$.

Flashcard 55: What is $\frac{1}{x+1}-\frac{1}{x-1}$ written as a single fraction?

Answer: $\frac{-2}{x^2-1}$. Use LCD $x^2-1$ to get $\frac{x-1-(x+1)}{x^2-1}$.

Flashcard 56: What is $\frac{x}{x^2-1}-\frac{1}{x+1}$ written as a single fraction?

Answer: $\frac{-1}{x^2-1}$. Rewrite $\frac{1}{x+1}$ with LCD $x^2-1$ then subtract.

Flashcard 57: What is $\frac{x^2-4}{x}\div\frac{x-2}{x+2}$ simplified?

Answer: $\frac{(x+2)^2}{x}$ (with $x \neq 0, 2, -2$). Multiply by reciprocal and simplify by canceling and factoring.

Flashcard 58: What is $\frac{2}{x-3}-\frac{x}{x-3}$ simplified?

Answer: $\frac{2-x}{x-3}$. Subtract numerators: $\frac{2-x}{x-3}$.

Flashcard 59: What is $\frac{a}{b}-\frac{c}{b}$ simplified, assuming $b\ne 0$?

Answer: $\frac{a-c}{b}$. Combine fractions with the same denominator.

Flashcard 60: What is $\frac{a}{b}\div\frac{c}{d}$ rewritten using multiplication?

Answer: $\frac{a}{b}\cdot\frac{d}{c}$ (require $c\ne 0$). Division by a fraction equals multiplication by its reciprocal.

Flashcard 61: What is $\frac{1}{x}-\frac{1}{x-1}$ written as a single fraction?

Answer: $\frac{-1}{x(x-1)}$. Use LCD $x(x-1)$ and combine: $\frac{x-1-x}{x(x-1)}$.

Flashcard 62: What is the LCD of $\frac{1}{(x-1)^2}$ and $\frac{1}{x-1}$?

Answer: $(x-1)^2$. Use the highest power of $(x-1)$ that appears.

Flashcard 63: What is the LCD of $\frac{1}{x^2-1}$ and $\frac{1}{x-1}$?

Answer: $x^2-1$ (since $x^2-1=(x-1)(x+1)$). Since $x^2-1$ contains $(x-1)$ as a factor, it's the LCD.

Flashcard 64: What is $\frac{x}{x-2}+\frac{2}{x-2}$ simplified?

Answer: $\frac{x+2}{x-2}$. Add numerators since denominators are the same.

Flashcard 65: What is $\frac{x}{5}\div\frac{x}{2}$ simplified (assume $x\ne 0$)?

Answer: $\frac{2}{5}$. Multiply by reciprocal and cancel: $\frac{x}{5}\cdot\frac{2}{x}$.

Flashcard 66: What is $\frac{3}{x+2}+\frac{1}{x+2}$ simplified?

Answer: $\frac{4}{x+2}$. Add numerators since denominators are identical.