Problem 13 - Elementary and Intermediate Algebra

Question

Add and simplify the result, if possible. \(\frac{9}{x}+\frac{2}{x}\)

Step-by-Step Solution

Answer

\(\frac{11}{x}\)

Step 1: Identify the Denominators

Both fractions in the expression \(\frac{9}{x}+\frac{2}{x}\) have the same denominator \(x\). This means you can combine them directly.

Step 2: Add the Numerators

Since the denominators are the same, you can add the numerators: \(9 + 2\). This gives \(\frac{9+2}{x}\).

Step 3: Simplify the Result

Simplify \(\frac{9+2}{x}\) by performing the addition in the numerator: \(9 + 2 = 11\). The simplified expression is \(\frac{11}{x}\).

Key Concepts

Adding Rational Expressions

To add rational expressions, you need to focus on the numerators and denominators of each term. A rational expression is basically a fraction with a polynomial in the numerator and/or denominator.
When adding these expressions, your main goal is to combine them into a single, simplified expression.

• If the denominators are already the same, you can directly add the numerators.
• If the denominators differ, you first need to find a common denominator, but more on that in the next section.

In our original problem, \[\frac{9}{x} + \frac{2}{x}\]both terms have the same denominator (x). This makes the process straightforward, as you can jump right into adding the numerators. After adding them, keep the common denominator unchanged, leading you to a single, combined rational expression.

Common Denominators

Finding a common denominator is a crucial step if the denominators of the rational expressions differ. The common denominator is usually the least common multiple of the denominators, designed so that each fraction can adjust without changing its value.

• Multiply each term by appropriate factors to make the denominators identical.
• Ensure that the process does not change the value of each individual expression.

In our task, since \[\frac{9}{x} + \frac{2}{x}\]already shares the same denominator (x), no further steps are necessary in this area. However, always check for and establish common denominators if they don't initially match—this will facilitate the combining of numerators in future problems.

Numerator Addition

Once you have determined that the denominators are either the same or have been made the same through finding common denominators, the next step is to add the numerators together. It's important to focus only on the numerators here, as the denominator will stay consistent.

• Add the numbers on top while keeping the denominator unchanged.
• If the numbers require simplification, do so to achieve the simplest form of the expression.

In the exercise\[\frac{9}{x} + \frac{2}{x}\]the numerators are 9 and 2. Adding these numbers gives us 11, leading to the expression:\[\frac{9+2}{x} = \frac{11}{x}\]This approach gives you the simplified form of the expression, illustrating a crucial step in adding rational expressions through combining like terms.