Problem 55
Question
Subtract: $$\frac{3}{x-4}-\frac{2}{x+2}$$
Step-by-Step Solution
Verified Answer
The result of the subtraction of the two given fractions is \( \frac{x+14}{(x-4)(x+2)} \).
1Step 1 - Find common denominator
Find the common denominator by multiplying the two denominators '\(x-4\)' and '\(x+2\)'. The common denominator is \( (x-4)(x+2) \).
2Step 2 - Rewrite the fractions over the common denominator
Rewrite each fraction over the common denominator. The first one is multiplied by '\(x+2\)' both in the numerator and denominator and becomes \( \frac{3(x+2)}{(x-4)(x+2)} \). The second one is multiplied by '\(x-4\)' both in the numerator and denominator and becomes \( \frac{2(x-4)}{(x-4)(x+2)} \).
3Step 3 - Subtract the fractions
Subtract the second fraction from the first one: \( \frac{3(x+2)}{(x-4)(x+2)} - \frac{2(x-4)}{(x-4)(x+2)} = \frac{3(x+2) - 2(x-4)}{(x-4)(x+2)} \).
4Step 4 - Simplify the Result
Simplify the numerator in the resulted fraction: \( \frac{3(x+2) - 2(x-4)}{(x-4)(x+2)} = \frac{3x+6 - 2x+8}{(x-4)(x+2)} = \frac{x+14}{(x-4)(x+2)} \).
Key Concepts
Common DenominatorSimplifying FractionsAlgebraic Expressions
Common Denominator
When subtracting rational expressions, finding a common denominator is a crucial step. Imagine trying to subtract fractions like \( \frac{3}{4} - \frac{2}{3} \). You can't simply subtract the numerators; you need a common base for those fractions. The same rule applies to rational expressions.
To find a common denominator, multiply each distinct denominator of the fractions. In our example, \( \frac{3}{x-4} \) and \( \frac{2}{x+2} \) have different denominators, \( x-4 \) and \( x+2 \). Multiply these to find the common denominator, resulting in \( (x-4)(x+2) \).
This approach ensures that both fractions have equivalent expression bases, allowing you to perform operations across them. Remember, a common denominator helps in not only subtracting but also adding fractions of various types.
To find a common denominator, multiply each distinct denominator of the fractions. In our example, \( \frac{3}{x-4} \) and \( \frac{2}{x+2} \) have different denominators, \( x-4 \) and \( x+2 \). Multiply these to find the common denominator, resulting in \( (x-4)(x+2) \).
This approach ensures that both fractions have equivalent expression bases, allowing you to perform operations across them. Remember, a common denominator helps in not only subtracting but also adding fractions of various types.
Simplifying Fractions
Simplifying fractions means making them as simple as possible. You achieve this by canceling out common factors from the numerator and denominator.
After subtracting the fractions, you might end up with a complex expression. Simplification makes it neater and easier to understand. In algebra, this involves expanding and combining like terms. For instance, simplifying \( \frac{3(x+2) - 2(x-4)}{(x-4)(x+2)} \) requires expanding the terms in the numerator first, providing \( \frac{3x + 6 - 2x + 8}{(x-4)(x+2)} \). By combining \( 3x \) and \( -2x \), you simplify this to \( x+14 \).
Remember to check if any further reduction is possible by factoring out and canceling with the denominator. Simplifying helps maintain a cleaner expression, making it easier to work with in future calculations.
After subtracting the fractions, you might end up with a complex expression. Simplification makes it neater and easier to understand. In algebra, this involves expanding and combining like terms. For instance, simplifying \( \frac{3(x+2) - 2(x-4)}{(x-4)(x+2)} \) requires expanding the terms in the numerator first, providing \( \frac{3x + 6 - 2x + 8}{(x-4)(x+2)} \). By combining \( 3x \) and \( -2x \), you simplify this to \( x+14 \).
Remember to check if any further reduction is possible by factoring out and canceling with the denominator. Simplifying helps maintain a cleaner expression, making it easier to work with in future calculations.
Algebraic Expressions
Algebraic expressions are combinations of variables and numbers using operations like addition, subtraction, multiplication, and division. They form the building blocks of more complex equations.
In the context of rational expressions, you're working with fractions where the numerator and/or the denominator is an algebraic expression. For example, \( \frac{3(x+2)}{(x-4)(x+2)} \) is a rational expression with algebraic components.
Handling algebraic expressions requires a good grasp of the basic algebraic rules, such as distribution (expanding expressions), combining like terms, and factoring. Developing a strong understanding of these rules is essential, especially when dealing with more complex algebraic equations. This ensures you can manipulate and work with expressions correctly, no matter how they're formatted.
In the context of rational expressions, you're working with fractions where the numerator and/or the denominator is an algebraic expression. For example, \( \frac{3(x+2)}{(x-4)(x+2)} \) is a rational expression with algebraic components.
Handling algebraic expressions requires a good grasp of the basic algebraic rules, such as distribution (expanding expressions), combining like terms, and factoring. Developing a strong understanding of these rules is essential, especially when dealing with more complex algebraic equations. This ensures you can manipulate and work with expressions correctly, no matter how they're formatted.
Other exercises in this chapter
Problem 54
make a rough sketch in a rectangular coordinate system of the graphs representing the equations in each system. The system, whose graphs are a line with negativ
View solution Problem 54
Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{aligned}x^{2}+y^{2} &
View solution Problem 55
Explain how to find the partial fraction decomposition of a rational expression with a repeated linear factor in the denominator.
View solution Problem 55
Application Exercises A planet's orbit follows a path described by \(16 x^{2}+4 y^{2}=64\) A comet follows the parabolic path \(y=x^{2}-4 .\) Where might the co
View solution