Detailed solutions to questions in How to Add, Subtract and Simplify Rational Expressions are presented.
Add, subtract and simplify the following with detailed solutions
\text{a)} \;\; \dfrac{7}{6} + \dfrac{1}{18} - \dfrac{5}{24}
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
\dfrac{7}{6} + \dfrac{1}{18} - \dfrac{5}{24} = \dfrac{7 \times \color{red}{12}}{6 \times \color{red}{12}} + \dfrac{1\times \color{red}{4}}{18\times \color{red}{4}} - \dfrac{5\times \color{red}{3}}{24\times \color{red}{3}}
and simplify as follows:
= \dfrac{84}{72} + \dfrac{4}{72} - \dfrac{15}{72} = \dfrac{84+4-15}{72} = \dfrac{73}{72}
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. |