Rational Equations

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What is a rational equation?

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A rational equation is an equation that involves at least one rational expression, which is a fraction where the numerator and/or denominator are polynomials.

Tags

CCSS.HSA.REI.A.2

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How do you solve a rational equation?

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To solve a rational equation, you typically find a common denominator, eliminate the fractions by multiplying both sides by that denominator, and then solve the resulting polynomial equation.

Tags

CCSS.HSA.REI.A.2

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What is the importance of checking for extraneous solutions in rational equations?

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Checking for extraneous solutions is important because multiplying both sides of a rational equation by a variable expression can introduce solutions that do not satisfy the original equation.

Tags

CCSS.HSA.REI.A.2

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What is the first step in solving the equation \( \frac{n}{n-4}+n=\frac{12-4n}{n-4} \)?

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The first step is to find a common denominator, which in this case is \( n-4 \), and then multiply through by this denominator to eliminate the fractions.

Tags

CCSS.HSA.REI.A.2

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What are the solutions to the equation \( \frac{n}{n-4}+n=\frac{12-4n}{n-4} \)?

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The solutions are \( n=-4 \) and \( n=3 \).

Tags

CCSS.HSA.REI.A.2

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How do you isolate a variable in a rational equation?

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To isolate a variable in a rational equation, you can cross-multiply if the equation is in the form of two fractions set equal to each other, or you can manipulate the equation algebraically to get the variable on one side.

Tags

CCSS.HSA.REI.A.2

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What is the solution to the equation \( \frac{5}{x+2}=\frac{1}{x-4} \)?

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The solution is \( x=5.5 \).

Tags

CCSS.HSA.REI.A.2

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What is the method to solve \( \frac{5}{3}-\frac{2}{x}=\frac{8}{x} \)?

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To solve this equation, you can first combine the fractions on the left side and then isolate \( x \) by cross-multiplying.

Tags

CCSS.HSA.REI.A.2

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What is the solution to the equation \( \frac{5}{3}-\frac{2}{x}=\frac{8}{x} \)?

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The solution is \( x=6 \).

Tags

CCSS.HSA.REI.A.2

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What is the general form of a rational expression?

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A rational expression is generally in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.

Tags

CCSS.HSA.APR.D.6