Solving Rational Expressions - Algebra II

The two fractions on the left side of the equation need a common denominator. We can easily do find one by multiplying both the top and bottom of each fraction by the denominator of the other.

becomes .

becomes .

Now add the two fractions:

To solve, multiply both sides of the equation by , yielding

.

Multiply both sides by 3:

Move all terms to the same side:

This looks like a complicated equation to factor, but luckily, the only factors of 37 are 37 and 1, so we are left with

.

Our solutions are therefore

and

.

Multiply both sides by :

Factor this using the -method. We split the middle term using two integers whose sum is and whose product is . These integers are :

Set each factor equal to 0 and solve separately:

or

Subtract 1 from both sides, then multiply all sides by :

A quadratic equation is yielded. We can factor the expression, then set each individual factor to 0.

Both of these solutions can be confirmed by substitution.

With rational equations we must first note the domain, which is all real numbers except . (If , then the term will be undefined.) Next, the least common denominator is , so we multiply every term by the LCD in order to cancel out the denominators. The resulting equation is . Subtract on both sides of the equation to collect all variables on one side: . Lastly, divide by the constant to isolate the variable, and the answer is . Be sure to double check that the solution is in the domain of our equation, which it is.

In order to subtract the fractions, multiply both denominators together in order to obtain the least common denominator.

Simplify the numerators.

Combine the numerators.

The answer is:

(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

To simplify this expression, we will need to multiply both denominators together to find the least common denominator.

Convert both fractions to the common denominator.

Combine the fractions.

The answer is:

First, simplify the expression before attempting to combine like-terms.

Combine like-terms.

The values can only be added or subtracted if there are like-terms in the expression. Since there are no like-terms in the question, the question is already simplified as is. All the other answers given are incorrect.

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

Therefore the answer is .

There are multiple operations required in this problem. The exponent must be eliminated before distributing the negative sign. Use the FOIL method which means to mulitply the first terms together, then multiply the outer terms together, then multiply the inner terms togethers, and lastly, mulitply the last terms together.

The negative sign can now be distributed.

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

First let us find a common denominator as follows:

Now we can subtract the numerators which gives us :

So the final answer is

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

With rational equations we must first note the domain, which is all real numbers except . (Recall, the denominator cannot equal zero. Thus, to find the domain set each denominator equal to zero and solve for what the variable cannot be.)

The least common denominator or and is . Multiply every term by the LCD to cancel out the denominators. The equation reduces to . We can FOIL to expand the equation to . Combine like terms and solve: . Factor the quadratic and set each factor equal to zero to obtain the solution, which is or . These answers are valid because they are in the domain.

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.

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:

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:

So the final answer is,

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:

First, find the common denominator, which is . Then, make sure to offset each numerator. Multiply by y to get . Multiply by x to get . Then, combine numerators to get . Then, put the numerator over the denominator to get your answer: .