To determine the difference of the rational expressions [tex]\(\frac{4c}{c+2}\)[/tex] and [tex]\(\frac{4c-7}{c-9}\)[/tex] with unlike denominators, follow these steps:
1. Identify the Least Common Denominator (LCD):
- The denominators for the given rational expressions are [tex]\( c+2 \)[/tex] and [tex]\( c-9 \)[/tex].
- Since these denominators are distinct and have no common factors, the least common denominator (LCD) is the product of both denominators: [tex]\( (c+2)(c-9) \)[/tex].
Thus, the statement should be:
The LCD is [tex]\( (c+2)(c-9) \)[/tex].
2. Create Equivalent Rational Expressions with a Common Denominator:
- For the rational expression [tex]\(\frac{4c}{c+2}\)[/tex]:
- To get the common denominator [tex]\( (c+2)(c-9) \)[/tex], multiply both the numerator and the denominator by the missing factor [tex]\( c-9 \)[/tex].
So, the statement becomes:
Both the numerator and denominator of the rational expression [tex]\(\frac{4c}{c+2}\)[/tex] are multiplied by [tex]\( c-9 \)[/tex] to create an equivalent rational expression with a common denominator.
- For the rational expression [tex]\(\frac{4c-7}{c-9}\)[/tex]:
- To get the common denominator [tex]\( (c+2)(c-9) \)[/tex], multiply both the numerator and the denominator by the missing factor [tex]\( c+2 \)[/tex].
Thus, the statement is:
Both the numerator and denominator of the rational expression [tex]\(\frac{4c-7}{c-9}\)[/tex] are multiplied by [tex]\( c+2 \)[/tex] to create an equivalent rational expression with a common denominator.
Putting it all together, the completed statements are:
1. The LCD is [tex]\((c+2)(c-9)\)[/tex].
2. Both the numerator and denominator of the rational expression [tex]\(\frac{4c}{c+2}\)[/tex] are multiplied by [tex]\( c-9 \)[/tex] to create an equivalent rational expression with a common denominator.
3. Both the numerator and denominator of the rational expression [tex]\(\frac{4c-7}{c-9}\)[/tex] are multiplied by [tex]\( c+2 \)[/tex] to create an equivalent rational expression with a common denominator.
