In Exercises 1–34, solve each rational equation. If an equation has no solution, so state. 2/(x+3) − 5/(x+1) = (3x+5)/(x²+4x+3)

Verified step by step guidance

1. Identify the least common denominator (LCD) of the rational expressions. The denominators are $x+3$, $x+1$, and $x^2+4x+3$. Notice that $x^2+4x+3$ factors into $(x+3)(x+1)$.
2. Multiply every term in the equation by the LCD, $(x+3)(x+1)$, to eliminate the fractions.
3. After multiplying, simplify each term. The first term becomes $2(x+1)$, the second term becomes $-5(x+3)$, and the right side becomes $3x+5$.
4. Combine like terms and simplify the equation to form a linear equation.
5. Solve the linear equation for $x$. Check your solution by substituting back into the original equation to ensure no division by zero occurs.

Key Concepts

Rational Equations

Rational equations are equations that involve fractions with polynomials in the numerator and denominator. To solve these equations, one typically finds a common denominator to eliminate the fractions, allowing for easier manipulation and simplification. Understanding how to work with rational expressions is crucial for solving these types of equations.

Finding Common Denominators

Finding a common denominator is essential when dealing with rational expressions, as it allows for the combination of fractions into a single expression. The common denominator is usually the least common multiple (LCM) of the individual denominators. This step is vital for simplifying the equation and solving for the variable.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This is particularly important in rational equations, as it helps to simplify expressions and identify potential solutions or restrictions on the variable, such as values that would make the denominator zero.