Problem 13 - Algebra 1: Concepts and Skills

Question

ADDING RATIONAL EXPRESSIONS. Simplify the expression. $$ \frac{2}{x+7}+\frac{5}{x+7} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{7}{x+7} \)

Step 1: Identify like denominators

In this case, the denominators of both fractions are the same and are \( x+7 \). This means these fractions can be added directly.

Step 2: Add the numerators

You can directly add the numerators of these fractions while keeping the denominator same. This gives \( \frac{2+5}{x+7} \)

Step 3: Simplify the expression

Adding the numerators in the fraction gives \( \frac{7}{x+7} \).

Key Concepts

Simplify Expressions

To simplify an expression means to make it as simple as possible. This usually involves combining like terms, reducing fractions, or factoring. In the context of rational expressions, simplification can be achieved by adding or subtracting terms with like denominators and then combining them into a single fraction.

For example, in the exercise \( \frac{2}{x+7}+\frac{5}{x+7} \), the process begins by observing that the denominators are the same, which leads us to simplify the expression by combining the numerators. When we add \(2\) and \(5\), we get \(7\), resulting in the simplified form \(\frac{7}{x+7}\). This simple reaction reduces the two separate fractions into a single fraction, making the expression more compact and easier to understand or further manipulate.

Like Denominators

In arithmetic with fractions, like denominators are the key to directly adding or subtracting rational expressions. When two rational expressions have the same denominator, we can perform addition or subtraction on the numerators alone.

Let's illustrate this with an example from the exercise: \(\frac{2}{x+7}+\frac{5}{x+7}\). Here, both fractions have \(x+7\) as their denominator. When denominators are the same, they remain fixed while the numerators undergo the arithmetic operation. This results in a new numerator which contains the sum or difference of the original numerators. Only when we have like denominators can we apply this straightforward method, otherwise, we would need to find a common denominator first.

Rational Expressions Arithmetic

Rational expressions arithmetic involves the addition, subtraction, multiplication, and division of fractions that contain polynomials in their numerators and denominators. Just like with numerical fractions, to add or subtract rational expressions, you need to have like denominators. Multiplication and division, on the other hand, do not require like denominators and have their own set of rules.

In the provided example, \(\frac{2}{x+7}+\frac{5}{x+7}\), we use arithmetic of addition, which is simplified by having like denominators. After adding the numerators and keeping the common denominator, the result, \(\frac{7}{x+7}\), is a simplified form of the original expressions. It's important to note that when dealing with more complex rational expressions, we may need to factor polynomials or find least common denominators to simplify the expressions before performing arithmetic operations.