# Algebra II : Adding and Subtracting Rational Expressions

## Example Questions

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### Example Question #1 : Simplifying Rational Expressions

Simplify

Explanation:

This is a more complicated form of

Find the least common denominator (LCD) and convert each fraction to the LCD, then add the numerators.  Simplify as needed.

which is equivalent to

Simplify to get

### Example Question #41 : Rational Expressions

Simplify:

Explanation:

Because the two rational expressions have the same denominator, we can simply add straight across the top.  The denominator stays the same.

### Example Question #42 : Rational Expressions

Simplify

The expression cannot be simplified.

Explanation:

a.  Find a common denominator by identifying the Least Common Multiple of both denominators.  The LCM of 3 and 1 is 3.  The LCM of  and  is . Therefore, the common denominator is .

b. Write an equivialent fraction to  using  as the denominator.  Multiply both the numerator and the denominator by  to get . Notice that the second fraction in the original expression already has  as a denominator, so it does not need to be converted.

The expression should now look like:

c. Subtract the numerators, putting the difference over the common denominator.

### Example Question #1 : Solving Rational Expressions

Combine the following expression into one fraction:

The two fractions cannot be combined as they have different denominators.

Explanation:

To combine fractions of different denominators, we must first find a common denominator between the two. We can do this by multiplying the first fraction by  and the second fraction by . We therefore obtain:

Since these fractions have the same denominators, we can now combine them, and our final answer is therefore:

### Example Question #41 : Rational Expressions

What is ?

Explanation:

We start by adjusting both terms to the same denominator which is 2 x 3 = 6

Then we adjust the numerators by multiplying x+1 by 2 and 2x-5 by 3

The results are:

### Example Question #45 : Rational Expressions

What is ?

Explanation:

Start by putting both equations at the same denominator.

2x+4 = (x+2) x 2 so we only need to adjust the first term:

Then we subtract the numerators, remembering to distribute the negative sign to all terms of the second fraction's numerator:

### Example Question #46 : Rational Expressions

Determine the value of .

Explanation:

(x+5)(x+3) is the common denominator for this problem making the numerators 7(x+3) and 8(x+5).

7(x+3)+8(x+5)= 7x+21+8x+40= 15x+61

A=61

### Example Question #1 : Solving Rational Expressions

Explanation:

First factor the denominators which gives us the following:

The two rational fractions have a common denominator hence they are like "like fractions".  Hence we get:

Simplifying gives us

### Example Question #1 : Solving Rational Expressions

Subtract:

Explanation:

First let us find a common denominator as follows:

Now we can subtract the numerators which gives us :

### Example Question #48 : Rational Expressions

Solve the rational equation:

or

or

no solution