Julius Export

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Wayne State University *

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2011

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Mathematics

Date

Apr 3, 2024

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docx

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Uploaded by KidMorning5563

User: (1) Given the expression fraction numerator 1 over denominator x minus 5 end fraction, choose three different values of x and evaluate this expression for those values. Is it possible to choose a value of x for which the value of the expression is greater than 10? If so, what is this value of x? If not, explain why it is not possible. (2) Explain the process of adding rational expressions with different denominators. Give an example. Julius: ### Part 1: Evaluating the Expression for Different Values of $x$ Given the expression: $$\frac{1}{x - 5}$$ Let's choose three different values for $x$ and evaluate the expression for those values. Common choices might be $x = 4$, $x = 6$, and $x = 10$. 1. For $x = 4$: $$\frac{1}{4 - 5} = \frac{1}{-1} = -1$$ 2. For $x = 6$: $$\frac{1}{6 - 5} = \frac{1}{1} = 1$$ 3. For $x = 10$: $$\frac{1}{10 - 5} = \frac{1}{5} = 0.2$$ To find if it's possible to choose a value of $x$ for which the value of the expression is greater than 10, we need to solve the inequality: $$\frac{1}{x - 5} > 10$$ This inequality can be solved by isolating $x$. Let's solve it: $$\frac{1}{x - 5} > 10$$ Multiplying both sides by $x - 5$ (assuming $x - 5 > 0$ to keep the inequality direction the same) and then dividing by 10 gives: $$1 > 10(x - 5)$$ $$\frac{1}{10} > x - 5$$ Adding 5 to both sides gives: $$x < 5 + \frac{1}{10}$$ $$x < 5.1$$ So, for the expression to be greater than 10, $x$ must be less than 5.1 but greater than 5 (since the expression is undefined at $x = 5$). A possible value of $x$ that satisfies this condition is $x = 5.05$, for example. ### Part 2: Adding Rational Expressions with Different Denominators To add rational expressions with different denominators, follow these steps: 1. Find a Common Denominator : The common denominator should be the least common multiple (LCM) of the denominators of the rational expressions. 2. Rewrite Each Fraction : Rewrite each fraction so that it has the common denominator. 3. Add the Numerators : Once the fractions have the same denominator, add their numerators. 4. Simplify the Result : If possible, simplify the resulting fraction. Example : Add the rational expressions $\frac{1}{x}$ and $\frac{2}{x+2}$. 1. The denominators are $x$ and $x+2$. The LCM of these denominators is $x(x+2)$, which will be our common denominator. 2. Rewrite each fraction: - $\frac{1}{x} = \frac{1(x+2)}{x(x+2)} = \frac{x+2} {x(x+2)}$ - $\frac{2}{x+2} = \frac{2x}{x(x+2)}$ 3. Add the numerators: - $\frac{x+2}{x(x+2)} + \ frac{2x}{x(x+2)} = \frac{x+2+2x}{x(x+2)} = \frac{3x+2}{x(x+2)}$ 4. The result, $\frac{3x+2}{x(x+2)}$, is already in its simplest form. This process allows you to add rational expressions with different denominators by first finding a common denominator and then combining the numerators accordingly.

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