Problem 40 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{2 b+3}{b+1}-\frac{b-4}{b+1} $$

Step-by-Step Solution

Answer

Question: Subtract the given rational expressions and simplify the result: $$\frac{2b+3}{b+1} - \frac{b-4}{b+1}$$ Answer: The simplified difference of the given rational expressions is: $$\frac{b + 7}{b + 1}$$.

Step 1: Identify the common denominator

In this case, the common denominator for both fractions is already given, which is (b+1).

Step 2: Subtract the numerators

Now, subtract the numerators of these fractions. It will look like this: $$ \frac{(2b+3)-(b-4)}{b+1} $$

Step 3: Simplify the numerator

Simplify the numerator by combining like terms: $$ \frac{2b + 3 - b + 4}{b+1} $$ $$ \frac{b + 7}{b+1} $$

Step 4: Write the final answer

The simplified rational expression is: $$ \frac{b + 7}{b + 1} $$

Key Concepts

Common Denominator

When working with subtraction of rational expressions, finding a common denominator is crucial. This is similar to finding a common ground for two fractions so that they can be compared or combined. In our example, we're lucky because the denominators are already the same, which is \(b+1\). But when they aren't, you'd often have to multiply the denominators together or find the least common multiple. This ensures that the fractions are speaking the same 'mathematical language' before we attempt to subtract them.

This common basis is key to working effectively with algebraic fractions. Without it, subtracting rational expressions would be like trying to subtract apples from oranges; it simply doesn't work without converting them to a single type of fruit, or in this case, a single denominator.

Combining Like Terms

Combining like terms is a method used to simplify an algebraic expression or equation. The term 'like terms' refers to terms that have the same variable raised to the same power. When subtracting rational expressions, after dealing with the common denominator, we combine like terms in the numerators.

In our example, combining \(2b\) and \(b\), and then \(3\) and \(4\), makes the expression neater and easier to handle. Always look for terms that can be combined, as it's one of the first steps toward simplifying an expression. It tidies up the math and often reveals a simpler structure hidden within the expression.

Simplifying Expressions

The process of simplifying expressions is fundamental in algebra. It's about making expressions easier to understand and work with. After subtracting and combining like terms, we simplify to make the expression as 'clean' as possible. For example, after the subtraction in our exercise, \(2b\) and \(b\) were combined to become \(b\), and \(3\) and \(4\) became \(7\), leaving us with the much simpler \(\frac{b + 7}{b + 1}\).

Simplification doesn't change the value of the expression; it simply presents it in a more digestible form. This step removes all unnecessary complexity, making it easier for anyone working with the expression to understand and utilize it.

Algebraic Fractions

An algebraic fraction is basically a fraction where the numerator, the denominator, or both are algebraic expressions. Here, our exercise involves algebraic fractions because they have variables (in this case \(b\)) in place of or in addition to numbers. Handling them requires all the techniques we've discussed—finding common denominators, combining like terms, and simplifying expressions.

Think of algebraic fractions as the bridge between arithmetic fractions and algebra. They obey the same rules as their numerical counterparts but they can solve broader sets of problems because they represent a whole range of possible values. Ending up with a simplified algebraic fraction like \(\frac{b + 7}{b + 1}\) is beneficial when you want to apply these values to different scenarios or solve for specific variables.