Add the rational expressions to find the sum. 5/x+2 + 6/x-3
To add the rational expressions, we need a common denominator. The common denominator in this case is (x+2)(x-3).
Now, we can rewrite the expressions using the common denominator:
5/(x+2) + 6/(x-3) = (5(x-3) + 6(x+2))/((x+2)(x-3))
Simplifying the numerator:
= (5x - 15 + 6x + 12)/((x+2)(x-3))
= (11x - 3)/((x+2)(x-3))
Therefore, the sum of the rational expressions is (11x - 3)/((x+2)(x-3)).
To add these rational expressions, we need to find a common denominator.
The denominators are x+2 and x-3. The common denominator would then be (x+2)(x-3).
Now, we can rewrite each fraction with the common denominator:
5/(x+2) = 5 * (x-3) / ((x+2)(x-3))
6/(x-3) = 6 * (x+2) / ((x+2)(x-3))
Now, we can add the fractions:
[5 * (x-3) / ((x+2)(x-3))] + [6 * (x+2) / ((x+2)(x-3))]
Next, we can simplify the expression:
[5x - 15 + 6x + 12] / ((x+2)(x-3))
Combining like terms in the numerator gives:
(11x - 3) / ((x+2)(x-3))
Therefore, the sum of the rational expressions 5/x+2 + 6/x-3 is (11x - 3) / ((x+2)(x-3)).
To add rational expressions, you need to have a common denominator.
Step 1: Find the least common multiple (LCM) of the denominators.
The denominators are x+2 and x-3. To find the LCM, factor each denominator completely:
x+2 = (x+2)
x-3 = (x-3)
The LCM is the product of the highest power of each factor. In this case, the LCM is (x+2)(x-3).
Step 2: Rewrite each rational expression with the common denominator.
To do this, multiply the numerator and denominator of each fraction by the factors that are missing in the other fraction.
For the first fraction, (5/(x+2)), we need to multiply the numerator and denominator by (x-3):
(5/(x+2)) * ((x-3)/(x-3)) = (5(x-3))/((x+2)(x-3))
For the second fraction, (6/(x-3)), we need to multiply the numerator and denominator by (x+2):
(6/(x-3)) * ((x+2)/(x+2)) = (6(x+2))/((x+2)(x-3))
Now both fractions have the common denominator of (x+2)(x-3).
Step 3: Add the numerators.
The numerators are (5(x-3)) and (6(x+2)).
Therefore, the sum of the two rational expressions is:
(5(x-3) + 6(x+2))/((x+2)(x-3))
Simplifying further, we distribute and combine like terms in the numerator:
(5x - 15 + 6x + 12)/((x+2)(x-3)) = (11x - 3)/((x+2)(x-3))
So, the sum of the given rational expressions is (11x - 3)/((x+2)(x-3)).