Table of Contents
Warmup Questions II
These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.
Warmup Question 2
Subtract $$\dfrac{1}{21}-\dfrac{3}{7}.$$
Show Answer 2
\begin{align*}&\dfrac{1}{21}-\dfrac{3}{7} \\=&\dfrac{1}{21}-\dfrac{9}{21}\\=& -\dfrac{8}{21}.\end{align*}
Warmup Question 3
Divide
$$\dfrac{6}{35}\div\dfrac{9}{14}.$$
Show Answer 3
\begin{align*} & \dfrac{6}{35}\div\dfrac{9}{14}\\ =& \dfrac{6}{35}\cdot \dfrac{14}{9}\\=&\dfrac{6\cdot 14}{35\cdot 9}\\=&\dfrac{2\cdot 3\cdot 2\cdot 7}{5\cdot 7\cdot 3\cdot 3}\\=&\dfrac{4}{15}\end{align*}
Warmup Question 4
Factor $$x^2-4x-21.$$
Show Answer 4
$$x^2-4x-21=(x-7)(x+3)$$
Quick Intro II
This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.
A Quick Intro to Rational Expressions & Addition and Subtraction of Rational Expressions & Multiplication and Division of Rational Expressions
Key Words. Rational Expression, numerator, denominator, simplifying/reducing, multiplication/division/addition/subtraction of rational expressions, LCD (least common denominator), like/unlike denominators, equivalent rational expressions
$\bigstar$ A rational expression is an expression of the form $\dfrac{p}{q}$, where $p$ and $q$ are polynomials with $p\neq 0$. In this case, $p$ is the numerator, and $q$ is the denominator.
$\bigstar$ The fraction $\dfrac{2}{6}$ (which is a rational expression!) can be simplified/reduced to $\dfrac{1}{3}$, because both the numerator $2$ and the denominator $6$ are multiple of $2$, so $2$ can be cancelled out:
$$\dfrac{2}{6}= \dfrac{1\cdot 2}{2\cdot 3} = \dfrac{1}{3}.$$
The same happens with rational expressions in general:
$$\dfrac{pr}{qr} = \dfrac{p}{q}.$$
This is the only rule that can be used to simplify/reduce rational expressions.
$\bigstar$ All the arithmetic operations with rational expressions are similar to the ones with fractions that you are already familiar with.
Multiplication: $$\dfrac{p}{q}\cdot \dfrac{r}{s}=\dfrac{pr}{qs}$$
Division: $$\dfrac{p}{q}\div\dfrac{r}{s}= \dfrac{p}{q}\cdot \dfrac{s}{r}=\dfrac{ps}{qr}$$
Addition and Subtraction with like denominators:
$$\dfrac{p}{q}+\dfrac{r}{q}=\dfrac{p+r}{q}$$
$$\dfrac{p}{q}-\dfrac{r}{q}=\dfrac{p-r}{q}$$
Addition and Subtraction with unlike denominators:
When the denominators are not the same, we need to factor them and find out what the LCD (least common denominator) is. Then multiply both numerator and the denominator so that the new denominators are all the same, namely, the LCD. The new rational expressions are equivalent to the original ones. Now we are in the case of like denominators.
Video Lesson II
Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!
Video Lesson – part 2
A description of the video
In this video you will see the following rational expressions:
- $\dfrac{40}{36}$
- $\dfrac{32x^2-4xy}{2xy}$
- $\dfrac{x^2-4}{x^2+x-6}$
- $\dfrac{x-1}{1-x}$
Video Lesson – part 3
A description of the video
In this video you will see the following rational expressions:
- $\dfrac{3}{35}\cdot \dfrac{21}{6}$
- $\dfrac{7}{15}\div \dfrac{21}{6}$
- $\dfrac{(x-2)}{(x-3)(x+2)}\cdot\dfrac{(x+1)(x-3)}{(x-2)}$
- $\dfrac{x^2-4}{x^2+x-6}\div\dfrac{x-2}{x^2+5x+6}$
- $\dfrac{2}{21}- \dfrac{9}{70}$
- $\dfrac{x}{2y}+ \dfrac{5y}{6z}$
- $\dfrac{1}{z-3}- \dfrac{1}{z}$
- $\dfrac{x}{x^2-5x+6}- \dfrac{x-1}{x^2-4}$
Try Questions II
Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.
Try Question 4
Simplify $$\dfrac{2a-4}{3a-6}.$$
Show Answer 4
\begin{align*} &\dfrac{2a-4}{3a-6}\\=&\dfrac{2(a-2)}{3(a-2)}\\=&\dfrac{2}{3}\end{align*}
Try Question 5
Divide
$$\dfrac{2}{2x-4}\div\dfrac{x-2}{4-x}.$$
Show Answer 5
\begin{align*} & \dfrac{2}{2x-4}\div\dfrac{x-2}{4-x}\\ =&\dfrac{2}{2x-4}\cdot\dfrac{4-x}{x-2}\\=&\dfrac{2(4-x)}{(2x-4)(x-2)}\\=&\dfrac{-2(x-4)}{2(x-2)(x-2)}\\= & -\;\dfrac{x-4}{x^2-4}\end{align*}
Try Question 6
Add $$\dfrac{4}{2x-1}+\dfrac{2-x}{1-2x}.$$
Show Answer 6
\begin{align*}&\dfrac{4}{2x-1}+\dfrac{2-x}{1-2x}\\=&\dfrac{4}{2x-1}-\dfrac{2-x}{2x-1}\\=&\dfrac{4-(2-x)}{2x-1}\\=&\dfrac{4-2+x}{2x-1}\\=&\dfrac{2+x}{2x-1}\end{align*}
WeBWorK II
You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.
WeBWorK II
It is time to do the homework on WeBWork:
ReducingRationalExpressions
AddRationalExpressions
AddRationalExpressions2
When you are done, come back to this page for the Exit Questions.
Exit Questions II
After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!
Exit Questions 2
- What is the LCD?
- When is the LCD used? Why?
- Why can’t $\dfrac{x+5}{x-5}$ be simplified?
$\bigstar$ Simplify the expression $$\dfrac{x-1}{x^2+x-6} -\dfrac{5}{3x-6}.$$
Show Answer 2
\begin{align*}& \dfrac{x-1}{x^2+x-6} -\dfrac{5}{3x-6}\\=&\dfrac{x-1}{(x-2)(x+3)} -\dfrac{5}{3(x-2)} \\& (\text{ the LCD is } 3(x-2)(x+3))\\=&\dfrac{3(x-1)}{3(x-2)(x+3)} -\dfrac{5(x+3)}{3(x-2)(x+3)}\\=&\dfrac{3x-3}{3(x-2)(x+3)} -\dfrac{5x+15}{3(x-2)(x+3)}\\=&\dfrac{3x-3-(5x+15)}{3(x-2)(x+3)} \\=&\dfrac{3x-3-5x-15}{3(x-2)(x+3)} \\=&\dfrac{-2x-18}{3(x-2)(x+3)} \\=&-\;\dfrac{2x(x+9)}{3(x-2)(x+3)} \end{align*}
