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Subtracting Rational Expressions With Different Denominators

Subtracting Rational Expressions With Different Denominators

The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.

Example: How to Subtract Rational Expressions with Different Denominators

Subtract: \(\frac{x}{x-3}-\frac{x-2}{x+3}.\)

Solution

The above image has 3 columns. It shows the steps on how to subtract rational expressions with different denominators for x divided by x minus three minus x plus x minus 3. Step 1 is to Determine if the expressions have a common denominator. Yes – go to step 2. No – Rewrite each rational expression with the LCD. Find the LCD. Rewrite each rational expression as an equivalent rational expression with the LCD. In the above expression, the answer is no. Find the LCD of x minus 3, x plus 3. To the right of this is x – 3: x – 3. Below that is x – 2: x – 2. A line is drawn. Below that is written the LCD is x – 3 times x plus 3. Rewrite as x times x plus 3 divided by x minus 3 times x plus 3 minus x minus 2 times x minus 3 divided by x plus 3 times x minus 3. Keep the denominators factored! Factor to get x squared plus 3 x divided by x minus 3 times x plus 3 minus x squared minus 5 x plus 6 divided by x minus 3 times x plus 3.Step 2 is to subtract the rational expressions. Subtract the numerators and place the difference over the common denominator to get x 2 plus 3 x minus x squared minus 5 x plus 6 divided by x minus 3 times x plus 3. Then to x squared plus 3 x minus x squared plus 5 x minus 6 divided by x minus 3 times x plus 3. Be careful with the signs! Then to 8 x minus 6 divided by x minus 3 times x plus 3.Step 3 is to simplify, if possible. The numerator and denominator have no factors in common. The answer is simplified to 2 times 4 x minus 3 divided by x minus 3 times x plus 3.

The steps to take to subtract rational expressions are listed below.

Subtract rational expressions.

  1. Determine if they have a common denominator.
    Yes – go to step 2.
    No – Rewrite each rational expression with the LCD.
    Find the LCD.
    Rewrite each rational expression as an equivalent rational expression with the LCD.
  2. Subtract the rational expressions.
  3. Simplify, if possible.

Example

Subtract: \(\frac{8y}{{y}^{2}-16}-\frac{4}{y-4}.\)

Solution

 .
Do the expressions have a common denominator? No.
Rewrite each expression with the LCD.
 
Find the LCD.  . 
Rewrite each rational expression as an equivalent rational expression with the LCD..
Simplify the numerators..
Subtract the rational expressions..
Simplify the numerators..
Factor the numerator to look for common factors..
Remove common factors..
Simplify..

There are lots of negative signs in the next example. Be extra careful!

Example

Subtract: \(\frac{-3n-9}{{n}^{2}+n-6}-\frac{n+3}{2-n}.\)

Solution

 .
Factor the denominator..
Since \(n-2\) and \(2-n\) are opposites, we will mutliply the second rational expression by\(\frac{-1}{-1}\)..
Simplify..
Do the expressions have a common denominator? No. 
Find the LCD.  . 
Rewrite each rational expression as an equivalent rational expression with the LCD..
Simplify the numerators..
Simplify the rational expressions..
Somplify the numerator..
Factor the numerator to look for common factors..
Simplify..

When one expression is not in fraction form, we can write it as a fraction with denominator 1.

Example

Subtract: \(\frac{5c+4}{c-2}-3.\)

Solution

 .
Write \(3\) as \(\frac{3}{1}\) to have 2 rational expressions..
Do the rational expressions have a common denominator? No. 
Find the LCD of \(c-2\) and \(1.\) LCD = \(c-2.\) 
Rewrite \(\frac{3}{1}\) as an equivalent rational expression with the LCD..
Simplify..
Subtract the rational expressions..
Simplify..
Factor to check for common factors..
There are no common factors; the rational expression is simplified. 

Add or subtract rational expressions.

  1. Determine if the expressions have a common denominator.
    Yes – go to step 2.
    No – Rewrite each rational expression with the LCD.
    Find the LCD.
    Rewrite each rational expression as an equivalent rational expression with the LCD.
  2. Add or subtract the rational expressions.
  3. Simplify, if possible.

We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example we will start by factoring all three denominators to find their LCD.

Example

Simplify: \(\frac{2u}{u-1}+\frac{1}{u}-\frac{2u-1}{{u}^{2}-u}.\)

Solution

 .
Do the rational expressions have a common denominator? No. 
Find the LCD.  . 
Rewrite each rational expression as an equivalent rational expression with the LCD..
 .
Write as one rational expression..
Simplify..
Factor the numerator, and remove common factors..
Simplify..

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