Solve each equation in Exercises 83–108 by the method of your choice. 1/x + 1/(x + 3) = 1/4

Verified step by step guidance

1. Start by finding a common denominator for the fractions on the left side of the equation. The common denominator for $\frac{1}{x}$ and $\frac{1}{x+3}$ is $x(x+3)$.
2. Rewrite each fraction with the common denominator: $\frac{1}{x}=\frac{x+3}{x(x+3)}$ and $\frac{1}{x+3}=\frac{x}{x(x+3)}$.
3. Combine the fractions on the left side: $\frac{x+3}{x(x+3)}+\frac{x}{x(x+3)}=\frac{2x+3}{x(x+3)}$.
4. Set the equation equal to the right side: $\frac{2x+3}{x(x+3)}=\frac{1}{4}$.
5. Cross-multiply to eliminate the fractions: $4(2x+3)=x(x+3)$.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions is crucial for solving equations involving them. In this case, the equation contains rational expressions that need to be combined or simplified to find the value of x.

Finding a Common Denominator

To solve equations involving multiple fractions, it is often necessary to find a common denominator. This allows you to combine the fractions into a single equation, making it easier to isolate the variable. In the given equation, the common denominator will help eliminate the fractions and simplify the solving process.

Cross-Multiplication

Cross-multiplication is a technique used to solve equations involving two fractions set equal to each other. By multiplying the numerator of one fraction by the denominator of the other, you can create a simpler equation without fractions. This method is particularly useful in this problem to eliminate the fractions and solve for x.