Add, Subtract and Simplify Rational Expressions
Questions With Detailed Solutions

Detailed solutions to questions in How to Add, Subtract and Simplify Rational Expressions are presented.

Add, Subtract and Simplify the Following Rational Expressions - Solutions

Question 1

Rational Expression Question 1
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
Solution to Rational Expression Question 1 Step 1
and simplify as follows:

Question 2

simplify expressions question 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.
solution to part 1 question b
We now expand, simplify and factor the numerator if possible .
solution to part 2 question b

Question 3

simplify expressions question 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.
solution to part 1 question c
The two rational expressions have the same denominator and they are added as follows:
solution to part 2 question c

Question 4

simplify expressions question 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 x² - 3x + 2 and x² + 2 x - 3 and find the LCM of Expressions.
x² - 3x + 2 = (x - 1) (x - 2)
x² + 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.
solution to part 1 question d
We now add the numerators expand and simplify.
solution to part 2 question d

Question 5

simplify expressions question e
Solution:
We rewrite the given expression with numerators and denominators in factored form and simplify if possible.
solution to part 1 question e
We cancel common factors.
solution to part 2 question e
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:
solution to part 3 question e
Expand and simplify.
solution to part 4 question e

Question 6

simplify expressions question 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 x² + 9 x - 5 and 2 x + 10 and find the LCM.
2x - 1 = 2x - 1
2 x² + 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.
solution to part 1 question f
We now add the numerators and simplify.
solution to part 2 question f

Question 7

simplify expressions question 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.
solution to part 1 question g
solution to part 2 question g
Expand and simplify.
solution to part 3 question g

Add, Subtract and Simplify Rational Expressions Questions With Detailed Solutions