In Exercises 33–68, add or subtract as indicated. 5/x + 3

Verified step by step guidance

1. Identify the terms in the expression: \(\frac{5}{x} + 3\). Notice that \(\frac{5}{x}\) is a rational expression and \(3\) is a constant.
2. To add these terms, express the constant \(3\) as a fraction with the same denominator as \(\frac{5}{x}\). Since the denominator is \(x\), rewrite \(3\) as \(\frac{3x}{x}\).
3. Now the expression becomes \(\frac{5}{x} + \frac{3x}{x}\), which allows you to combine the fractions because they have a common denominator.
4. Add the numerators while keeping the denominator the same: \(\frac{5 + 3x}{x}\).
5. The simplified expression is \(\frac{5 + 3x}{x}\). This is the result of adding \(\frac{5}{x}\) and \(3\).

Key Concepts

Adding and Subtracting Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator and denominator. To add or subtract them, you must have a common denominator. Once the denominators match, you combine the numerators accordingly and simplify the result if possible.

Finding the Least Common Denominator (LCD)

The least common denominator is the smallest expression that both denominators can divide into without a remainder. Finding the LCD allows you to rewrite each rational expression with the same denominator, enabling addition or subtraction of the numerators.

Simplifying Rational Expressions

After combining rational expressions, simplifying involves factoring numerators and denominators and canceling common factors. This process reduces the expression to its simplest form, making it easier to interpret and use in further calculations.