8.4 Add and subtract rational expressions with unlike denominators (Page 2/4)

Elementary algebra Page 2 / 4

Add: $\frac{7}{12} + \frac{5}{18} .$

Solution


Find the LCD of 12 and 18.
Rewrite each fraction as an equivalent fraction with the LCD.
Add the fractions.
The fraction cannot be simplified.

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Add: $\frac{11}{30} + \frac{7}{12} .$

$\frac{57}{30}$

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Add: $\frac{3}{8} + \frac{9}{20} .$

$\frac{33}{40}$

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Now we will add rational expressions whose denominators are monomials.

Add: $\frac{5}{12 x^{2} y} + \frac{4}{21 x y^{2}} .$

Solution


Find the LCD of $12 x^{2} y$ and $21 x y^{2}$ .

Rewrite each rational expression as an equivalent fraction with the LCD.
Simplify.
Add the rational expressions.
There are no factors common to the numerator and denominator. The fraction cannot be simplified.

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Add: $\frac{2}{15 a^{2} b} + \frac{5}{6 a b^{2}} .$

$\frac{4 b + 25 a}{30 a^{2} b^{2}}$

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Add: $\frac{5}{16 c} + \frac{3}{8 c d^{2}} .$

$\frac{5 d^{2} + 6}{16 c d^{2}}$

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Now we are ready to tackle polynomial denominators.

How to add rational expressions with different denominators

Add: $\frac{3}{x - 3} + \frac{2}{x - 2} .$

Solution

$The above image shows the steps to add fractions whose denominators are monomials for the example 5 divided by 12 x squared y plus 4 divided by 21 x y squared. Find the LCD of 12 x squared y and 21 x y squared. To the right of this expression is 12 x squared y equals 2 times 2 times 3 times x times x times y. Below that is 21 x y squared equals 3 times 7 times x times y times y. A line is drawn. Below that is LCD equals 2 times 2 times 3 times 7 times x times x times y times y. Below that is LCD equals 84 x squared y squared. Rewrite each rational expression as an equivalent fraction with the LCD. The original equation is shown. Below that is 5 times 7 y divided by 12 x squared y times 7 y plus 4 times 4 x divided by 21 x y squared times 4 x. Simplify to get 35 y divided by 84 x squared y squared plus 16 x divided by x squared y squared. Add the rational expressions 16 x plus 35 y divided by 84 x squared y squared. There are no factors common to the numeration and denominator. The fraction cannot be simplified.$ Step 2 is to add the rational expression. Then, add the numerators and place the sum over the common denominator to get 3 x minus 6 plus 2 x minus 6 divided by x minus 3 times x minus 2.

Step 2 is to add the rational expression. Then, add the numerators and place the sum over the common denominator to get 3 x minus 6 plus 2 x minus 6 divided by x minus 3 times x minus 2.

Step 3 is to simplify, if possible. Because 5 x minus 12 cannot be factored, the answer is simplified to 5 x minus 12 divided by x minus 3 times x minus 2.

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Add: $\frac{2}{x - 2} + \frac{5}{x + 3} .$

$\frac{7 x - 4}{(x + 3) (x - 2)}$

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Add: $\frac{4}{m + 3} + \frac{3}{m + 4} .$

$\frac{7 m + 25}{(m + 3) (m + 4)}$

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The steps to use to add rational expressions are summarized in the following procedure box.

Add rational expressions.

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.
Add the rational expressions.
Simplify, if possible.

Add: $\frac{2 a}{2 a b + b^{2}} + \frac{3 a}{4 a^{2} - b^{2}} .$

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.
Add the rational expressions.
Simplify the numerator.
Factor the numerator.
There are no factors common to the numerator and denominator. The fraction cannot be simplified.

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Add: $\frac{5 x}{x y - y^{2}} + \frac{2 x}{x^{2} + y^{2}} .$

$\frac{x (5 x + 7 y)}{y (x - y) (x + y)}$

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Add: $\frac{7}{2 m + 6} + \frac{4}{m^{2} + 4 m + 3} .$

$\frac{7 m + 15}{2 (m + 3) (m + 1)}$

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Avoid the temptation to simplify too soon! In the example above, we must leave the first rational expression as $\frac{2 a (2 a - b)}{b (2 a + b) (2 a - b)}$ to be able to add it to $\frac{3 a \cdot b}{(2 a + b) (2 a - b) \cdot b}$ . Simplify only after you have combined the numerators.

Add: $\frac{8}{x^{2} - 2 x - 3} + \frac{3 x}{x^{2} + 4 x + 3} .$

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 fraction with the LCD.
Simplify the numerators.
Add the rational expressions.
Simplify the numerator.
The numerator is prime, so there are no common factors.

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Add: $\frac{1}{m^{2} - m - 2} + \frac{5 m}{m^{2} + 3 m + 2} .$

$\frac{5 m^{2} - 9 m + 2}{(m - 2) (m + 1) (m + 2)}$

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Add: $\frac{2 n}{n^{2} - 3 n - 10} + \frac{6}{n^{2} + 5 n + 6} .$

$\frac{2 (n^{2} + 6 n - 15)}{(n + 2) (n - 5) (n + 3)}$

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Subtract 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.

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.

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Subtract: $\frac{y}{y + 4} - \frac{y - 2}{y - 5} .$

$\frac{−7 y + 8}{(y + 4) (y - 5)}$

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Subtract: $\frac{z + 3}{z + 2} - \frac{z}{z + 3} .$

$\frac{4 z + 9}{(z + 2) (z + 3)}$

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

Subtract rational expressions.

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.
Subtract the rational expressions.
Simplify, if possible.

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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