What is the lowest common denominator of $\frac{x-5}{x^2-1}-1=-\frac{2 x}{x-1}$?
A. $x^2+1$
B. $x^3+1$
C. $x^3-x^2-x+1$
D. $x^2-1$
Answer (1)
Factor the denominators: x 2 − 1 = ( x − 1 ) ( x + 1 ) .
Identify the unique factors: ( x − 1 ) and ( x + 1 ) .
Multiply the unique factors to find the LCD: ( x − 1 ) ( x + 1 ) = x 2 − 1 .
The lowest common denominator is x 2 − 1 .
Explanation
Identify the Denominators We are asked to find the lowest common denominator (LCD) of the equation x 2 − 1 x − 5 − 1 = − x − 1 2 x . The denominators in the equation are x 2 − 1 and x − 1 . The LCD is the smallest expression that is divisible by each of the denominators.
Factor the Denominators To find the LCD, we first factor each denominator. The denominator x 2 − 1 can be factored as a difference of squares: x 2 − 1 = ( x − 1 ) ( x + 1 ) . The other denominator is x − 1 , which is already in simplest form.
Determine the LCD Now we identify the unique factors present in the denominators. We have the factors ( x − 1 ) and ( x + 1 ) . The LCD is the product of these unique factors, each raised to the highest power that appears in any of the denominators. In this case, ( x − 1 ) appears with a power of 1 in both denominators, and ( x + 1 ) appears with a power of 1 in the first denominator. Therefore, the LCD is ( x − 1 ) ( x + 1 ) = x 2 − 1
Select the Correct Option Comparing our result with the given options, we see that the lowest common denominator is x 2 − 1 , which corresponds to option D.
Examples
When solving equations involving fractions, finding the lowest common denominator is crucial for combining the fractions and simplifying the equation. For example, if you are calculating the total resistance in a parallel circuit with resistors having values that are expressed as fractions, you need to find the LCD to simplify the expression and find the total resistance. Similarly, in mixture problems, where you combine different solutions with varying concentrations, the LCD helps in determining the final concentration of the mixture.
