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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Adding fractions using the Least Common Denominator

Next we multiply these together. Here's the result of the third step:

The result of this multiplication is the LCD. Multiplied out, we get 360.
So, the LCD of and is

• Having the LCD in factored form is very useful. We will see why when we actually

Let's try this with our previous example:

This time, we will find and use the LCD.

Step 1:

Step 2: 2, 3, 5

Step 3:. (The highest power of each factor is only 1,
which we don't need to write.)
Now multiply each factor together:

The LCD is . (That's the end of the LCD process.)

Next we convert the fractions to the same denominator. We will make use of the fact
that the denominators and the LCD are already factored. To convert to a fraction
with the LCD in the bottom, compare the factorization of to the .
The LCD has a factor of 5 that 6 doesn't have (all other factors of 6 and the LCD are the
same). So we multiply the fraction (top and bottom) by 5:

Next compare (the denominator of ) to the .
The LCD has a "2" that 15 doesn't have. So we multiply top and bottom by 2:

Now we've converted the fractions; the last step is to add them together:

Notice that the answer we get, , can still be reduced:.

The numbers we had to work with during the addition were smaller than the numbers
we had to consider the first time we did this problem; and in both cases, we still had to
factor in order to simplify the fraction. But by using the LCD, we only had to factor
smaller numbers.

Here's an example where we find the LCD of three fractions:

Ex: . Find the LCD of all three fractions (we won't do the addition).

Step 1:

Step 2: 2, 3, 7

Step 3: . (Each factor raised to its highest power.) Then multiply:

The LCD is 504. (Finished!)

Finally, notice that the LCD of two or more fractions is always the smallest number that
can be evenly divided by the denominators of the fractions. For example, 504 is divisible
by each of the numbers 8, 28, and 18; and no smaller number can be divided by all
three. In other words, the LCD is also the least common multiple (LCM) of the
denominators of the fractions. The LCD
of and is 30; the LCM of the numbers 6 and 15 is also 30.

You should try all of the following:

EXERCISES.

 Find the least common multiple of the given numbers using the LCD process. In problems 6 - 10, add or subtract the fractions by converting to the LCD.

Some thoughts on studying mathematics and being successful, or
"The Eleven Steps to a Good Grade In Math"

1) Take good, complete class notes, employing the lecture outline as a guide.

2) Plan to spend about two hours on each assignment. You may sometimes need to
spend more.

3) As soon after the class section as possible, study your class notes and examples. You
may even want to rewrite your notes for greater clarity. (This rewrite process serves
to make you begin thinking about the ideas, the first step in learning.)

4) Then read carefully the section(s) of the textbook related to the material, giving
special attention to the vocabulary, basic concepts and problem examples. You may
want to relate this reading to your class notes, highlighting the concepts which have
been discussed.

5) Now begin the assignment. (To do so earlier is foolhardy, since the assignment is
designed to be practice and application of the ideas and concepts, which you really
need to grasp and understand before you do the assignment; hence, prior study of
the ideas is essential!)

6) Check your answers in the back of the textbook as you go along. If you are having
difficulties, go back and restudy, looking at the examples done in class and in the

7) If problems arise at this point, and you are unable to work them out for yourself or
with one another (group study is encouraged - we can learn from each other's
problems and analyses), then the time has come to seek help. Attend the problem
session
before the next class, or if that is impossible, seek help directly form the
instructor during office hours. Do not wait until the next class to ask the question;
there may not be time, or we may not be able to get to your problem.

8) When you have successfully completed your assignment, go back and reread the
text sections and class notes
again, for emphasis and review of the concepts. This
won’t take very long, but it pays big dividends in locking in the concepts, processes,
and terminology into your "memory bank."

9) Successful students find that the next step in doing one's assignment is to look
at what is coming up in the next class; you can easily see what section we will
be discussing by looking at your syllabus. Do a quick read over of the appropriate
text sections, not for mastery but for general overview. Get familiar with the new
terms and basic concepts, so when they are presented in class they wont seem so
"foreign" to you.

10) Finally, successful students also have found that, just prior to the next class (perhaps
fifteen minutes before class begins, or even on the way to class) a quick reread of
the "old" class notes
as well as the "new" text section gives them an edge, especially
if there is a quiz (announced or unannounced).

11) One final suggestion: don't get discouraged, and above all don't give up too easily.
Mathematics is challenging, but it can also be fun, interesting, intellectually
rewarding, and very useful!