Question

Find the LCD for StartFraction 8 Over x minus c EndFraction and StartFraction 4 Over c minus x EndFraction
.

Answer 1

To find the least common denominator (LCD) of the fractions \(\frac{8}{x - c}\) and \(\frac{4}{c - x}\), we start by examining the denominators.

1. The first fraction has the denominator \(x - c\).
2. The second fraction has the denominator \(c - x\).

Next, notice that \(c - x\) can be rewritten as:

\[
c - x = -(x - c)
\]

So, to compare the two denominators directly, we recognize that \(c - x\) is the negative of \(x - c\).

Now, the least common denominator (LCD) will be the positive version of the absolute value of one of the denominators. Since both denominators represent the same rational expression in terms of magnitude, the LCD of the two fractions is simply:

\[
\text{LCD} = |x - c|
\]

Thus, the least common denominator for the fractions \(\frac{8}{x - c}\) and \(\frac{4}{c - x}\) is:

\[
|x - c|
\]

Or, if you prefer to represent it without absolute values in a context where the sign does not matter (such as when substituting back), you could express the LCD directly as:

\[
x - c
\]

It's important to note when working with these expressions that \(x - c\) can be a negative quantity if \(x < c\), so be careful with subsequent calculations.