Add, subtract and simplify the following with detailed solutions
a)
Solution:
The three denominators in the fractions above are different and therefore we need to find a common denominator.
We first find the lowest common multiple (LCM) of the two denominators 6, 18 and 24.
6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 80,...
18: 18, 36, 54, 72, 90,...
24: 24, 48, 72, 96...
The lowest common denominator is 72 and we now convert all 3 denominators to the common denominator 72 and simplify as follows:
b)
Solution:
The two rational expressions have different denominators. In order to add the rational expressions above, we need to convert them to a common denominator. The two denominators x + 5 and x + 2 have no common factors hence their LCM is given by:
LCM = (x + 5)(x + 2)
We now use the LCM as the common denominator and rewrite the rational expressions with the same denominator as follows.
We now expand, simplify and factor the numerator if possible .
c)
Solution:
In order to add a rational expression with an expression without denominator, we convert the one without denominator into a rational expression then add them.
The two rational expressions have the same denominator and they are added as follows:
d)
Solution:
The two rational expressions have different denominators. In order to add the rational expressions above, we need to convert them to a common denominator. We first factor completely the two denominators x^{ 2}  3x + 2 and x^{ 2} + 2 x  3 and find the LCM of Expressions.
x^{ 2}  3x + 2 = (x  1) (x  2)
x^{ 2} + 2 x  3 = (x  1)(x + 3)
LCM = (x  1)(x  2)(x + 3)
We now use the LCM as the common denominator and rewrite the rational expressions with the same denominator as follows.
We now add the numerators expand and simplify.
e)
Solution:
We rewrite the given expression with numerators and denominators in factored form and simplify if possible.
We cancel common factors.
The two denominators x + 1 and x + 3 have no common factors and therefore their LCD is (x + 1)(x + 3). We rewrite the above with the common factor (x + 1)(x + 3) as follows:
Expand and simplify.
f)
Solution:
The three rational expressions have different denominators. In order to subtract/add the rational expressions above, we need to convert them to a common denominator.List and factor completely the three denominators 2x  1 , 2 x^{ 2} + 9 x  5 and 2 x + 10 and find the LCM.
2x  1 = 2x  1
2 x^{ 2} + 9 x  5 = (2x  1)(x + 5)
2x+10 = 2(x + 5)
LCM = 2(2x  1)(x + 5)
We now use the LCM as the common denominator and rewrite the rational expressions with the same denominator as follows.
We now add the numerators and simplify.
g)
Solution:
The two rational expressions have different denominators. In order to subtract/add the rational expressions above, we need to convert them to a common denominator. List and factor completely the two denominators y(x y  y + 3 x  3) and 2 x  2 and find the LCM.
y(x y  y + 3 x  3) = y( y(x  1) + 3 (x  1)) = y(x  1)(y + 3)
2 x  2 = 2(x  1)
LCM = 2 y (x  1)(y + 3)
We now use the LCM as the common denominator and rewrite the rational expressions with the same denominator as follows.
Expand and simplify.

