Add, Subtract, Simplify Rational Expressions Examples With Solutions
Examples on adding, subtracting and simplifying rational expressions for Grade 11 are presented along with detailed solutions and more questions with detailed Solutions and explanations are included.
We first start with adding, subtracting and simplifying fractions and then we move on to rational expressions.
An online calculator to simplify rational expressions is included and may be used to check results.
Adding, subtracting and simplifying rational expressions is done in the same way as adding, subtracting and simplifying fractions. Two cases are possible:
case 1: The fractions or rational expressions have the same denominator, we therefore add or subtract as follows:
case 2: The fractions or rational expressions do not have the same denominator, we first convert to a common denominator then add or subtract.
Integers only are included in In fractions, while algebraic expressions are included in rational expressions.
If you have difficulties in adding, subtracting and simplifying fractions and rational expressions, this tutorial will help you overcome those difficulties on the condition that you understand every step involved in solving these questions and also spend more time practicing if needed. I will present the examples below starting with fractions first and then with rational expressions, with more challenging questions as you walk through the tutorial. You need to understand each step!
Example 1
Subtract and simplify:
\[
\dfrac{2}{3} - \dfrac{4}{3}
\]
Solution:
The two fractions have the same denominator and therefore we subtract as follows
\[
\dfrac{2}{3} - \dfrac{4}{3} = \dfrac{2-4}{3} = - \dfrac{2}{3}
\]
Example 2
Subtract and simplify:
\[
\dfrac{1}{5} - \dfrac{7}{10}
\]
Solution:
The two fractions have different denominators and we therefore need to convert them to the same denominator.
We first find the lowest common multiple (LCM) of the two denominators 5 and 10.
5: 5, 10, 15, ... (multiply 5 by 1, 2, 3, ... to obtain a list of multiples of 5)
10: 10, 20, 30, ... (multiply 10 by 1, 2, 3, ... to obtain a list of multiples of 10)
The first common multiple (or the lowest, in red in the lists above) will be used as the common denominator which is also called lowest common denominator (LCD).
We now convert all denominator to the common denominator 10 as follows:
\(
\dfrac{1}{5} - \dfrac{7}{10} = \dfrac{1 \times \color{red}{2}}{5 \times \color{red}{2}} - \dfrac{7}{10} = \dfrac{2}{10} - \dfrac{7}{10}
\)
then simplify
\(
= \dfrac{2-7}{10} = - \dfrac{5}{10} = - \dfrac{1}{2}
\)
Example 3
Simplify:
\[
\dfrac{5}{8} + \dfrac{1}{12} - \dfrac{5}{16}
\]
Solution:
The three denominators are different and therefore we need to find a common denominator.
We first find the lowest common multiple (LCM) of the two denominators 8, 12 and 16.
8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80,...
12: 12, 24, 36, 48, 60, 72, 84, 96,...
16: 16, 32, 48, 64, 80, 96...
The lowest common denominator is 48 and we now convert all 3 denominators to the common denominator 48.
\(
\dfrac{5}{8} + \dfrac{1}{12} - \dfrac{5}{16} = \dfrac{5 \times \color{red}{6}}{8 \times \color{red}{6}} + \dfrac{1 \times \color{red}{4}}{12 \times \color{red}{4}} - \dfrac{5 \times \color{red}{3}}{16 \times \color{red}{3}}
\)
and then simplify as follows:
\(
= \dfrac{30}{48} + \dfrac{4}{48} - \dfrac{15}{48} = \dfrac{30+4-15}{48} = \dfrac{19}{48}
\)
Add, Subtract and Simplify Rational Expressions
Example 4
Subtract and simplify:
\[
\dfrac{2-x}{x+2} - \dfrac{4}{x+2}
\]
Solution:
The two rational expressions have the same denominator and therefore we subtract the numerators and keep the same denominator as follows
\( \require{cancel} \)
\(
\dfrac{2-x}{x+2} - \dfrac{4}{x+2} = \dfrac{2-x-4}{x+2}
\)
then simplify
\(
\dfrac{-x-2}{x+2} = \dfrac{-\cancel{(x+2)}}{\cancel{x+2}} = - 1 , \text {for } x \ne -2
\)
Example 5
Write as a rational expression:
\[ \dfrac{1}{x + 5} + x - 3 \]
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:
Expand the product (x - 3)(x + 5) and simplify.
NOTE: For the following examples, you need to know How to Find lowest common multiple (LCM) of Expressions and also practice on questions on detailed solutions on LCM.
Example 6
Add and simplify: .
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 3x + 6 and x + 2 and find the LCM.
3 x + 6 = 3(x + 2)
x + 2 = x + 2
LCM = 3(x + 2)
We now use the LCM as the common denominator and rewrite the rational expressions with the same denominator as follows.
We now add and simplify.
Example 7
Add and simplify: .
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 - x - 2 and x 2 + 4 x + 3 and find the LCM.
x 2 - x - 2 = (x + 1)(x - 2)
x 2 + 4 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, factor x + 2 and simplify.
The numerator in factored form is very useful in many situations: solving rational inequalities, solving rational equations, graphing rational functions, ....
case 2: The fractions or rational expressions do not have the same denominator, we first convert to a common denominator then add or subtract.
.
.
