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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Simplifying Fractions

## Examples with solutions

Example 1:

Simplify .

solution:

There is no possibility of factoring the denominator further here, since neither term in the denominator is a product. The numerator is a trinomial, and so possibly can be factored into a product of two binomials:

As explained earlier in these notes, for this to be possible, we would need to find numbers ‘a’ and ‘b’ such that a + b = -5 and ab = 6. You have probably already noticed that a = -2 and b = -3 will work. It is easy to confirm that

(x – 3)(x – 2) = x 2 – 5x + 6.

So, our original fraction becomes

in simplest form.

Example 2:

Simplify .

solution:

Proceeding as in the previous example, we quickly find that

At first, it may not appear that any cancellation is possible because neither of the factors in the numerator look the same as the denominator. However, recall that (x – 5) = -(5 – x). So, we can write our fraction here as

since now the presence of the common factor (5 – x) in the numerator and denominator is obvious. Cancelling that common factor leads to the final result.

Example 3:

Simplify .

solution:

By now, you are very familiar with the process: factor the numerator and denominator and cancel common factors. Here this gives

Example 4:

Simplify .

solution:

Factoring and taking into account for the minus sign in the denominator gives

as the final simplified result.