Operating with Rational Expressions - Algebra
Card 1 of 30
What is $\frac{1}{x+1}+\frac{1}{x-1}$ written as a single fraction?
What is $\frac{1}{x+1}+\frac{1}{x-1}$ written as a single fraction?
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$\frac{2x}{x^2-1}$. Use LCD $ (x+1)(x-1)=x^2-1 $ to get $\frac{x-1+x+1}{x^2-1}$
$\frac{2x}{x^2-1}$. Use LCD $ (x+1)(x-1)=x^2-1 $ to get $\frac{x-1+x+1}{x^2-1}$
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What is $\frac{x}{x+2}\cdot\frac{x+2}{3}$ simplified?
What is $\frac{x}{x+2}\cdot\frac{x+2}{3}$ simplified?
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$\frac{x}{3}$ (with $x\ne -2$). The $(x+2)$ terms cancel, leaving $\frac{x}{3}$.
$\frac{x}{3}$ (with $x\ne -2$). The $(x+2)$ terms cancel, leaving $\frac{x}{3}$.
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What does it mean for rational expressions to be closed under addition and subtraction?
What does it mean for rational expressions to be closed under addition and subtraction?
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The sum or difference of two rational expressions is a rational expression. Adding or subtracting rational expressions yields another rational expression.
The sum or difference of two rational expressions is a rational expression. Adding or subtracting rational expressions yields another rational expression.
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What does it mean for rational expressions to be closed under multiplication?
What does it mean for rational expressions to be closed under multiplication?
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The product of two rational expressions is a rational expression. Multiplying rational expressions produces another rational expression.
The product of two rational expressions is a rational expression. Multiplying rational expressions produces another rational expression.
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What condition is required for closure under division of rational expressions?
What condition is required for closure under division of rational expressions?
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Division is allowed only by a nonzero rational expression. Cannot divide by zero, so the divisor must be nonzero.
Division is allowed only by a nonzero rational expression. Cannot divide by zero, so the divisor must be nonzero.
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What is $\frac{x^2-1}{x+1}\div(x-1)$ simplified?
What is $\frac{x^2-1}{x+1}\div(x-1)$ simplified?
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$1$ (with $x\neq -1$ and $x\neq 1$). Rewrite division as multiplication by reciprocal, then simplify.
$1$ (with $x\neq -1$ and $x\neq 1$). Rewrite division as multiplication by reciprocal, then simplify.
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What is $\frac{1}{x-3}-\frac{1}{x+3}$ written as a single fraction?
What is $\frac{1}{x-3}-\frac{1}{x+3}$ written as a single fraction?
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$\frac{6}{x^2-9}$. Use LCD $x^2-9$ to get $\frac{x+3-(x-3)}{x^2-9}=\frac{6}{x^2-9}$.
$\frac{6}{x^2-9}$. Use LCD $x^2-9$ to get $\frac{x+3-(x-3)}{x^2-9}=\frac{6}{x^2-9}$.
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What is the reciprocal of the rational expression $\frac{a}{b}$ (with $a\ne 0$ and $b\ne 0$)?
What is the reciprocal of the rational expression $\frac{a}{b}$ (with $a\ne 0$ and $b\ne 0$)?
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$\frac{b}{a}$. Flip the numerator and denominator to get the reciprocal.
$\frac{b}{a}$. Flip the numerator and denominator to get the reciprocal.
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What is $\frac{4}{x^2-9}+\frac{1}{x+3}$ written as a single fraction?
What is $\frac{4}{x^2-9}+\frac{1}{x+3}$ written as a single fraction?
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$\frac{x+7}{x^2-9}$. Use LCD $x^2-9=(x+3)(x-3)$ to combine the fractions.
$\frac{x+7}{x^2-9}$. Use LCD $x^2-9=(x+3)(x-3)$ to combine the fractions.
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What is $\frac{2}{x-3}-\frac{x}{x-3}$ simplified?
What is $\frac{2}{x-3}-\frac{x}{x-3}$ simplified?
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$\frac{2-x}{x-3}$. Subtract numerators: $\frac{2-x}{x-3}$.
$\frac{2-x}{x-3}$. Subtract numerators: $\frac{2-x}{x-3}$.
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What is $\frac{x}{x-2}+\frac{2}{x-2}$ simplified?
What is $\frac{x}{x-2}+\frac{2}{x-2}$ simplified?
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$\frac{x+2}{x-2}$. Add numerators since denominators are the same.
$\frac{x+2}{x-2}$. Add numerators since denominators are the same.
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What is $\frac{1}{x+1}-\frac{1}{x-1}$ written as a single fraction?
What is $\frac{1}{x+1}-\frac{1}{x-1}$ written as a single fraction?
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$\frac{-2}{x^2-1}$. Use LCD $x^2-1$ to get $\frac{x-1-(x+1)}{x^2-1}$.
$\frac{-2}{x^2-1}$. Use LCD $x^2-1$ to get $\frac{x-1-(x+1)}{x^2-1}$.
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What is $\frac{2}{x}-\frac{1}{2x}$ written as a single fraction?
What is $\frac{2}{x}-\frac{1}{2x}$ written as a single fraction?
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$\frac{3}{2x}$. Use LCD $2x$: $\frac{4}{2x}-\frac{1}{2x}=\frac{3}{2x}$.
$\frac{3}{2x}$. Use LCD $2x$: $\frac{4}{2x}-\frac{1}{2x}=\frac{3}{2x}$.
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What is the LCD of $\frac{1}{x^2-1}$ and $\frac{1}{x-1}$?
What is the LCD of $\frac{1}{x^2-1}$ and $\frac{1}{x-1}$?
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$x^2-1$ (since $x^2-1=(x-1)(x+1)$). Since $x^2-1$ contains $(x-1)$ as a factor, it's the LCD.
$x^2-1$ (since $x^2-1=(x-1)(x+1)$). Since $x^2-1$ contains $(x-1)$ as a factor, it's the LCD.
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What is the LCD of $\frac{1}{(x-1)^2}$ and $\frac{1}{x-1}$?
What is the LCD of $\frac{1}{(x-1)^2}$ and $\frac{1}{x-1}$?
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$(x-1)^2$. Use the highest power of $(x-1)$ that appears.
$(x-1)^2$. Use the highest power of $(x-1)$ that appears.
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Identify the missing condition for $\frac{a}{b}\div\frac{c}{d}$ to be defined.
Identify the missing condition for $\frac{a}{b}\div\frac{c}{d}$ to be defined.
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$\frac{c}{d}\ne 0$ (equivalently $c\ne 0$ and $d\ne 0$). The divisor must be nonzero for division to be defined.
$\frac{c}{d}\ne 0$ (equivalently $c\ne 0$ and $d\ne 0$). The divisor must be nonzero for division to be defined.
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What is $\frac{2}{x}+\frac{3}{x}$ simplified?
What is $\frac{2}{x}+\frac{3}{x}$ simplified?
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$\frac{5}{x}$. Add numerators when denominators are the same.
$\frac{5}{x}$. Add numerators when denominators are the same.
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What is $\frac{a}{b}\div\frac{c}{d}$ rewritten using multiplication?
What is $\frac{a}{b}\div\frac{c}{d}$ rewritten using multiplication?
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$\frac{a}{b}\cdot\frac{d}{c}$ (require $c\ne 0$). Division by a fraction equals multiplication by its reciprocal.
$\frac{a}{b}\cdot\frac{d}{c}$ (require $c\ne 0$). Division by a fraction equals multiplication by its reciprocal.
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What is $\frac{a}{b}\cdot\frac{c}{d}$ simplified in one step?
What is $\frac{a}{b}\cdot\frac{c}{d}$ simplified in one step?
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$\frac{ac}{bd}$ (with $b\ne 0$ and $d\ne 0$). Multiply numerators together and denominators together.
$\frac{ac}{bd}$ (with $b\ne 0$ and $d\ne 0$). Multiply numerators together and denominators together.
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What is $\frac{a}{b}-\frac{c}{b}$ simplified, assuming $b\ne 0$?
What is $\frac{a}{b}-\frac{c}{b}$ simplified, assuming $b\ne 0$?
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$\frac{a-c}{b}$. Combine fractions with the same denominator.
$\frac{a-c}{b}$. Combine fractions with the same denominator.
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What restriction must be stated for a rational expression $\frac{p(x)}{q(x)}$?
What restriction must be stated for a rational expression $\frac{p(x)}{q(x)}$?
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$q(x)\ne 0$ (exclude values that make the denominator $0$). The denominator cannot equal zero to avoid undefined expressions.
$q(x)\ne 0$ (exclude values that make the denominator $0$). The denominator cannot equal zero to avoid undefined expressions.
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What is $\frac{a}{b}+\frac{c}{b}$ simplified, assuming $b\ne 0$?
What is $\frac{a}{b}+\frac{c}{b}$ simplified, assuming $b\ne 0$?
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$\frac{a+c}{b}$. Combine fractions with the same denominator.
$\frac{a+c}{b}$. Combine fractions with the same denominator.
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What is the domain restriction for $\frac{1}{x-3}$?
What is the domain restriction for $\frac{1}{x-3}$?
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$x\ne 3$. Setting the denominator equal to zero: $x-3=0$ gives $x=3$.
$x\ne 3$. Setting the denominator equal to zero: $x-3=0$ gives $x=3$.
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What is $\frac{1}{x}\div\frac{1}{x-3}$ simplified?
What is $\frac{1}{x}\div\frac{1}{x-3}$ simplified?
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$\frac{x-3}{x}$ (with $x\ne 0$ and $x\ne 3$). Multiply by reciprocal: $\frac{1}{x}\cdot\frac{x-3}{1}$.
$\frac{x-3}{x}$ (with $x\ne 0$ and $x\ne 3$). Multiply by reciprocal: $\frac{1}{x}\cdot\frac{x-3}{1}$.
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What is $\frac{x}{5}\div\frac{x}{2}$ simplified (assume $x\ne 0$)?
What is $\frac{x}{5}\div\frac{x}{2}$ simplified (assume $x\ne 0$)?
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$\frac{2}{5}$. Multiply by reciprocal and cancel: $\frac{x}{5}\cdot\frac{2}{x}$.
$\frac{2}{5}$. Multiply by reciprocal and cancel: $\frac{x}{5}\cdot\frac{2}{x}$.
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What is $\frac{x^2-4}{x-2}$ simplified?
What is $\frac{x^2-4}{x-2}$ simplified?
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$x+2$ (with $x \neq 2$). Factor $x^2-4=(x-2)(x+2)$ and cancel $(x-2)$.
$x+2$ (with $x \neq 2$). Factor $x^2-4=(x-2)(x+2)$ and cancel $(x-2)$.
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What is the rule for dividing rational expressions $\frac{a}{b}\div\frac{c}{d}$?
What is the rule for dividing rational expressions $\frac{a}{b}\div\frac{c}{d}$?
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$\frac{a}{b}\cdot\frac{d}{c}$ with $b\ne 0$ and $c\ne 0$ and $d\ne 0$. Division becomes multiplication by the reciprocal of the divisor.
$\frac{a}{b}\cdot\frac{d}{c}$ with $b\ne 0$ and $c\ne 0$ and $d\ne 0$. Division becomes multiplication by the reciprocal of the divisor.
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What is the quotient $\frac{2}{x}\div\frac{3}{x+1}$ simplified?
What is the quotient $\frac{2}{x}\div\frac{3}{x+1}$ simplified?
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$\frac{2(x+1)}{3x}$. Multiply by reciprocal: $\frac{2}{x}\cdot\frac{x+1}{3}$.
$\frac{2(x+1)}{3x}$. Multiply by reciprocal: $\frac{2}{x}\cdot\frac{x+1}{3}$.
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What is $\frac{3x^2y}{6xy^2}$ simplified?
What is $\frac{3x^2y}{6xy^2}$ simplified?
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$\frac{x}{2y}$ (with $x\ne 0$ and $y\ne 0$). Cancel common factors: $3x$ and $y$.
$\frac{x}{2y}$ (with $x\ne 0$ and $y\ne 0$). Cancel common factors: $3x$ and $y$.
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What is $\frac{2x}{x^2}$ simplified?
What is $\frac{2x}{x^2}$ simplified?
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$\frac{2}{x}$ (with $x\ne 0$). Cancel one factor of $x$ from numerator and denominator.
$\frac{2}{x}$ (with $x\ne 0$). Cancel one factor of $x$ from numerator and denominator.
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