Problem 50 - Algebra 1

Question

Find the LCD of \(\frac{-2}{x+9}\) and \(\frac{5 x}{x^{2}+9 x}\) (A) \(\frac{x-1}{(x-1)(2 x+1)}\) \((\mathbf{B})-\frac{x}{x-1}\) (c) \(\frac{2 x^{2}+1}{(x-1)(2 x+1)}\) (D) \(\frac{2 x^{2}-1}{(x-1)(2 x+1)}\)

Step-by-Step Solution

Answer

None of the given options corresponds to the LCD of the given fractions

Step 1: Identify the Denominators

The denominators of the two fractions are \(x + 9\) and \(x^2 + 9x\). For the second denominator, factor out the common term to get \(x(x+9)\).

Step 2: Find the LCD

The LCD is the expression that includes all individual factors of the denominators. Since the denominators share the same factors of \(x\) and \(x+9\), the LCD is actually \(x(x+9)\).

Step 3: Compare with Choices

We find that none of the options matches the LCD derived from the given fractions. The problem seems to be misstated or it could be that we are given incorrect options.

Key Concepts

Denominator

In any fraction, the denominator is the bottom number, showing into how many parts the whole is divided. For example, in the fraction \( \frac{-2}{x+9} \), the denominator is \(x + 9\). The denominator of a fraction is crucial because it tells you the size of each part.

• The denominator determines the value of each fraction relative to the whole.
• In algebraic terms, it can include variables and constants.
• To operate on two fractions, their denominators need to have a common base, which is why finding a least common denominator (LCD) is necessary.

In the original exercise, understanding what the denominators are is the first step before proceeding to find the least common denominator for addition or subtraction. It is essential to identify all factors contained in each denominator to find the LCD effectively.

Factoring

Factoring is the process of breaking down an expression into simpler 'factors' that, when multiplied together, give the original expression. It's a critical skill when working with algebraic fractions, especially for finding the least common denominator.

• This process often involves recognizing patterns like the difference of squares, perfect square trinomials, or common factors.
• For example, in the denominator \(x^2 + 9x\), factoring helps reveal \(x(x+9)\).
• Factoring simplifies expressions to make further operations easier and more understandable.

In this exercise, factoring the second fraction's denominator illustrates the common process of simplifying an algebraic expression to identify all factors of the denominator.

Common Factors

Common factors are elements that two or more numbers or expressions share. Identifying these is key when simplifying expressions or finding least common denominators.

• Common factors can be numerical values, variables, or expressions like \(x+9\).
• In the denominators \(x + 9\) and \(x(x+9)\), the common factor is \(x+9\).
• Recognizing common factors helps streamline calculations by minimizing the number of terms.

When determining the LCD, identifying these common factors ensures that each expression fully incorporates the factors from all denominators, enabling effective comparison and calculation.