Problem 64 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x+3}{x^{2}-x-30}+\frac{x-2}{30+x-x^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{x+5}{x^{2}-x-30}\)

Step 1: Rearrange the Denominator of the Second Fraction

Rearrange the denominator of the second fraction to match it with the first one. For this, the expression will become \( \frac{2 x+3}{x^{2}-x-30} - \frac{x-2}{x^{2}-x-30}\)

Step 2: Add the Fractions

Now, add the two fractions. Since the denominators are the same, you can easily combine the numerators and keep the denominator the same. This results in \( \frac{2x+3-(x-2)}{x^{2}-x-30}\) which simplifies to \( \frac{x+5}{x^{2}-x-30}\)

Step 3: Simplify the Fraction

Finally, simplify the fraction if possible. In this case, the fraction \( \frac{x+5}{x^{2}-x-30}\) cannot be further simplified, so it is the final result.

Key Concepts

Additive Inverses

Additive inverses are a fundamental concept in algebra, playing a crucial role in solving equations and simplifying expressions. An additive inverse of a number is essentially "what you add to a number to make it zero." When you have number \( d \), its additive inverse is \( -d \). When you add them together, you get zero: \( d + (-d) = 0 \). This property is incredibly useful when you need to rearrange or simplify algebraic expressions.

In the context of fractions or algebraic expressions, identifying terms or denominators that are additive inverses can simplify operations like addition or subtraction. Consider two expressions where their denominators are additive inverses. Such rearrangement can help simplify the terms, as evidenced in the exercise where rearranging the denominators helped in simplifying the fractions.​

Fractions

Understanding fractions is essential in algebra, especially when dealing with expressions that need addition or subtraction. A fraction consists of a numerator (top part) and a denominator (bottom part). In the given exercise, both fractions already had similar denominators. This similarity made it straightforward to add them without finding another common denominator.

When dealing with similar denominators, you can directly add or subtract the numerators. For example:

• If the fractions are \( \frac{a}{b} + \frac{c}{b} \), you can combine them as \( \frac{a+c}{b} \).
• For subtraction, \( \frac{a}{b} - \frac{c}{b} \) becomes \( \frac{a-c}{b} \).

In some cases, you might need to manipulate or rearrange expressions to have similar denominators, just like in the exercise, where rearranging created a matching denominator to facilitate addition.

Simplifying Expressions

Simplifying expressions is a process of making an algebraic expression as simple as possible by performing all possible operations. It makes working with algebra easier and more effective. In our exercise, simplifying started once the fractions were combined. It's essential to:

• Combine like terms in the numerators.
• Check if you can factor or reduce expressions, although sometimes further simplification isn't possible, as in \( \frac{x+5}{x^2-x-30} \).

Simplification allows you to see the more refined form of an expression, which can be easier to understand and use for further operations or problem-solving. Always double-check if parentheses or factoring can simplify things further, ensuring the expression is in its simplest form.