Finding the Least Common Denominator (LCD) for fractions with algebraic expressions involves identifying a common expression that both denominators can multiply into. Here's how I found the LCD in the equation $\frac{2x}{x + 1} + \frac{5}{2x} = 2$:

Step 1: Identify the Denominators

The two fractions in the equation are:

• $\frac{2x}{x + 1}$ with denominator $x + 1$
• $\frac{5}{2x}$ with denominator $2x$

Step 2: Determine the LCD

To find the LCD, we need to find the smallest expression that both $x + 1$ and $2x$ can divide into.

The LCD must include each factor from both denominators:

• The first denominator is $x + 1$, so $x + 1$ must be part of the LCD.
• The second denominator is $2x$, so $2x$ must also be part of the LCD.

Thus, the LCD is the product of these two expressions:

$\text{LCD} = (x + 1)(2x)$

Step 3: Use the LCD to Combine the Fractions

With the LCD $(x + 1)(2x)$, you can rewrite each fraction by multiplying both the numerator and denominator by whatever is missing from the original denominators to reach the LCD.

For example:

• For $\frac{2x}{x + 1}$, multiply by $\frac{2x}{2x}$ to get $\frac{2x \cdot 2x}{(x + 1)(2x)}$.
• For $\frac{5}{2x}$, multiply by $\frac{x + 1}{x + 1}$ to get $\frac{5(x + 1)}{(2x)(x + 1)}$.

Once both fractions are expressed with the LCD as the denominator, you can add or subtract them as needed.

Would you like more examples or further clarification on finding LCDs?

Related Questions:

1. How do you simplify fractions with algebraic expressions after finding the LCD?
2. What are the common pitfalls when finding the LCD in algebraic expressions?
3. How can you factor expressions to simplify the process of finding an LCD?
4. When is it necessary to find an LCD in algebraic equations?
5. How do you verify that the LCD has been correctly determined?

Tip:

Always factor the denominators first when possible; this makes finding the LCD easier and helps avoid mistakes.