Question

Identify the least common denominator of the equation. $$\frac{2}{x+1}+\frac{x}{x+4}=\frac{1}{2}$$

Short answer

The least common denominator is \(2*(x+1)*(x+4)\).

Step-by-step solution

Identify the denominators

The denominators in the equation are \(x+1\), \(x+4\), and \(2\).

Find the least common denominator among the three

Since all the denominators are different and they are not factors of each other, the least common denominator of these three is simply the product of all the terms i.e., \(2*(x+1)*(x+4)\).

Verify the least common denominator

Quickly check the LCD. All the original denominators \(2\), \(x+1\), \(x+4\) should divide into the LCD. Indeed, \(2*(x+1)*(x+4)\) is divisible by all of them, confirming it as the least common denominator.

Key concepts

Rational Expressions

Rational expressions are similar to fractions, but instead of just numbers, they include polynomials in their numerators and denominators. A rational expression is a quotient of two polynomials. For example, \(\frac{2}{x+1}\) is a rational expression where "2" is the numerator and \(x+1\) is the denominator. These types of expressions are a fundamental part of algebra.Understanding rational expressions is important because they appear frequently in algebraic problems, especially when solving equations or simplifying expressions. To work effectively with rational expressions, it is crucial to set common denominators when adding or subtracting them, just as you would with regular fractions.
This lets you combine the terms without changing their values. In our original exercise, finding the least common denominator (LCD) allows us to combine the expressions \(\frac{2}{x+1}\) and \(\frac{x}{x+4}\), which is a key step in solving the equation.

Algebraic Fractions

Algebraic fractions are a special type of rational expressions where the numerator and/or the denominator are polynomials. Just like normal fractions, you can perform operations such as addition, subtraction, multiplication, and division on them.When dealing with algebraic fractions, identifying a common denominator is crucial for performing addition or subtraction. Here, the denominators are \(x+1\), \(x+4\), and \(2\). Since these are not multiples of each other, the LCD is the product of all these terms, \(2(x+1)(x+4)\).
Having a common denominator simplifies the process of solving algebraic problems, since it allows us to combine terms in a meaningful way. This technique helps us rewrite algebraic fractions so they can be easily manipulated and solved.
When the denominator is composed of different expressions, multiplying them together ensures the new denominator can accommodate each term individually.

Solving Equations

Solving equations involving rational expressions often requires you to find the least common denominator first. This permits you to combine or eliminate terms effectively. In our exercise, to solve the given equation \(\frac{2}{x+1}+\frac{x}{x+4}=\frac{1}{2}\), the strategy is to multiply through by the LCD, \(2(x+1)(x+4)\).
By eliminating the denominators, you transform the rational expressions into simpler algebraic expressions. This step is crucial because it reduces a complex fraction equation into a polynomial equation, which is generally easier to solve.After multiplying by the LCD, you'll get an equation without fractions, making it much simpler to handle. Solving such equations may require expanding brackets, combining like terms, or using algebraic techniques to isolate the variable.Finally, solving these transformed equations reveals the value of the unknown, providing the solution to the original problem. This process highlights the importance of correctly identifying and using the least common denominator in algebra.