Problem 47
Question
For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x+3}+\frac{2}{x-2}}{\frac{2}{x+3}-\frac{3}{x-2}} $$
Step-by-Step Solution
Verified Answer
The simplified form is \(\frac{3x+4}{-x-13}\).
1Step 1 - Simplify the Numerator
First, find a common denominator for the fractions in the numerator, which are \(\frac{1}{x+3}\) and \(\frac{2}{x-2}\). The common denominator is \((x+3)(x-2)\). Rewrite the numerator: \(\frac{1 \times (x-2) + 2 \times (x+3)}{(x+3)(x-2)} = \frac{x-2 + 2x + 6}{(x+3)(x-2)} = \frac{3x+4}{(x+3)(x-2)}\)
2Step 2 - Simplify the Denominator
Similarly, simplify the denominator fractions \(\frac{2}{x+3}\) and \(\frac{3}{x-2}\) by finding a common denominator: \(\frac{2 \times (x-2) - 3 \times (x+3)}{(x+3)(x-2)}= \frac{2x-4 - 3x - 9}{(x+3)(x-2)} = \frac{-x-13}{(x+3)(x-2)}\)
3Step 3 - Combine the Simplified Fractions
Now, divide the simplified numerator by the simplified denominator: \(\frac{\frac{3x+4}{(x+3)(x-2)}}{\frac{-x-13}{(x+3)(x-2)}} = \frac{3x+4}{-x-13}\). Remove the common denominator terms as they cancel each other out.
4Step 4 - Simplify the Division
The final expression is \(\frac{3x+4}{-x-13}\). This is the simplified form of the given complex fraction.
Key Concepts
Understanding Common DenominatorsSimplifying Rational ExpressionsWorking with Fractions
Understanding Common Denominators
To simplify or evaluate a fraction with other fractions in the numerator and denominator, you first need to find a common denominator. A common denominator is a shared multiple of the denominators of the fractions you are working with. This helps combine the fractions into a single fraction, making them easier to work with. For example, if you have fractions with denominators of \(x+3\) and \(x-2\), then their common denominator would be \((x+3)(x-2)\). By finding this common denominator, we can rewrite each fraction so they share this denominator, allowing us to add or subtract them easily.
This step is crucial for simplifying fractions and makes the subsequent arithmetic operations straightforward.
This step is crucial for simplifying fractions and makes the subsequent arithmetic operations straightforward.
Simplifying Rational Expressions
Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying these expressions involves a few steps: finding common denominators, combining fractions, and reducing the resulting fraction if possible.
Let's break it down: Suppose you have the expression: \(\frac{\frac{1}{x+3} + \frac{2}{x-2}}{\frac{2}{x+3} - \frac{3}{x-2}} \). You first focus on simplifying the numerator and the denominator separately by finding their common denominators and combining them. This results in a single fraction for the numerator and another single fraction for the denominator. After you have simplified both, you can then divide one by the other. Remember, dividing by a fraction is the same as multiplying by its reciprocal.
This step-by-step simplification ensures that you keep your work organized and accurate.
Let's break it down: Suppose you have the expression: \(\frac{\frac{1}{x+3} + \frac{2}{x-2}}{\frac{2}{x+3} - \frac{3}{x-2}} \). You first focus on simplifying the numerator and the denominator separately by finding their common denominators and combining them. This results in a single fraction for the numerator and another single fraction for the denominator. After you have simplified both, you can then divide one by the other. Remember, dividing by a fraction is the same as multiplying by its reciprocal.
This step-by-step simplification ensures that you keep your work organized and accurate.
Working with Fractions
Understanding fractions is fundamental in algebra and other areas of mathematics. A fraction represents a part of a whole and consists of a numerator (a top part) and a denominator (a bottom part). When working with multiple fractions, combining them often requires finding a common denominator. This ensures the fractions can be added, subtracted, or compared properly. For example, \(\frac{1}{3} + \frac{1}{4} \) would need the common denominator of 12 to be combined: \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\).
Understanding operations involving fractions is key to solving more complex problems, like the given exercise. When fractions have variables in the denominators, the common-denominator approach remains the same. This ensures algebraic expressions are combined correctly and simplified properly.
Always break down complex fractions into manageable steps to avoid mistakes and simplify your solution process.
Understanding operations involving fractions is key to solving more complex problems, like the given exercise. When fractions have variables in the denominators, the common-denominator approach remains the same. This ensures algebraic expressions are combined correctly and simplified properly.
Always break down complex fractions into manageable steps to avoid mistakes and simplify your solution process.
Other exercises in this chapter
Problem 46
For exercises 39-82, simplify. $$ \frac{h}{5 k} \div \frac{h}{10 m} $$
View solution Problem 47
For exercises \(45-48\), the formula \(R=\frac{U F}{P}\) describes the glomular filtration rate by a kidney \(R\). Is the relationship of the given variables a
View solution Problem 47
For exercises \(5-48\), simplify. $$ \frac{2 v^{2}}{2 v^{2}+5 v-12}+\frac{13 v}{2 v^{2}+5 v-12}-\frac{24}{2 v^{2}+5 v-12} $$
View solution Problem 47
For exercises 39-82, simplify. $$ \frac{8 a b}{21 c^{2}} \div \frac{2 a^{2}}{3 c} $$
View solution