Solving Rational Equations (Proportions)

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What are the domain restrictions for rational equations?

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The domain restrictions for rational equations are the values of x that make the denominator equal to zero. For example, if the equation has a denominator of (x-2)(x+1), then the domain restrictions would be x ≠ 2 and x ≠ -1.

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How do you solve a rational equation?

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To solve a rational equation, first find a common denominator, then multiply both sides of the equation by that denominator to eliminate the fractions. After that, solve the resulting equation for x.

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What is a proportion in the context of rational equations?

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A proportion is an equation that states that two ratios are equal. In rational equations, proportions can be used to set up equations that can be solved for unknown variables.

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What does it mean to simplify a rational expression?

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Simplifying a rational expression means reducing it to its simplest form by canceling common factors in the numerator and denominator.

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What is the first step in solving the equation \( \frac{x + 1}{10} = \frac{2}{5} \)?

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The first step is to cross-multiply to eliminate the fractions, which gives you (x + 1) * 5 = 2 * 10.

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What is the solution to the equation \( \frac{x + 1}{10} = \frac{2}{5} \)?

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The solution is x = 3.

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What is the significance of checking your solutions in rational equations?

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Checking your solutions is important to ensure that they do not violate any domain restrictions, as solutions that make the denominator zero are not valid.

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How do you find the least common denominator (LCD) for rational expressions?

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To find the least common denominator, identify the denominators of all fractions involved, factor them, and then take the highest power of each factor present.

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What is the process for solving a proportion?

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To solve a proportion, cross-multiply the two ratios and then solve the resulting equation for the unknown variable.

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What happens if you try to solve a rational equation with an invalid solution?

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If you try to solve a rational equation with an invalid solution, it will lead to a contradiction or an undefined expression, indicating that the solution is not acceptable.