Solving Rational Expressions - Algebra II

Begin by cross-multiplying.

Distribute the on the left side and expand the polynomial on the right.

Combine like terms and rearrange to set the equation equal to zero.

Now we can isolate and solve for by adding to both sides.

First we convert each of the denominators into an LCD which gives us the following:

Now we add or subtract the numerators which gives us:

Simplifying the above equation gives us the answer which is:

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.

With rational equations we must first note the domain, which is all real numbers except and . That is, these are the values of that will cause the equation to be undefined. Since the least common denominator of , , and is , we can mulitply each term by the LCD to cancel out the denominators and reduce the equation to . Combining like terms, we end up with . Dividing both sides of the equation by the constant, we obtain an answer of . However, this solution is NOT in the domain. Thus, there is NO SOLUTION because is an extraneous answer.

Since the two fractions already have a common denominator, you can add the fractions by adding up the two numerators and keeping the common denominator:

Next you will algebraically solve for by isolating it on one side of the equation. The first step is to multiply each side by :

Cancel out the on the left and distribute out on the right. Then solve for by subtracting to the left and subtracting 10 to the right. Finally divide each side by negative 2:

To solve this rational equation, start by cross multiplying:

Then, distribute the right side:

Finally, subtract from both sides and bring the over to the left side:

Dividing by gives the answer:

The first step is to multiply everything by a common denominator. One way to do this is to multiply the entire equation by all three denominators:

Then, to solve for , use the quadratic formula:

The first step of problems like this is to try to factor the quadratic and see if it shares a factor with the linear polynomial in the denominator. And as it turns out,

So our rational function is equal to

which is as simplified as it can get.

When dividing fractions, we multiply by the reciprocal of the second fraction. Therefore, the problem becomes:

Our final unfactored expression is therefore .

In order to solve for , we need to consider this equation as two proportions being set equal:

Now, we can cross-multiply.

Distribute the on the right side:

Subtract from both sides to start combining like terms:

Now, this is just a two-step equation. Start by subtracting from both sides:

Divide by .

First, cross-multiply:

Once we simplify we are left with the following two quadratic equations:

In order to solve a quadratic, we need to have it equal to zero and then we can use the quadratic formula. What we need to do now is combine like terms. We can subtract all of the terms on the left from the like terms on the right:

This gives us:

Now we can use the quadratic formula:

Where the quadratic equation follows the pattern:

Therefore, we can use the terms in our quadratic equation and rewrite the equation as follows:

Since we can re-write as and , our answer becomes

Consider this problem as 2 proportions set equal:

Now, we can cross-multiply.

Using FOIL (method of simplifying binomials by multiplying following the pattern of first terms, outer terms, inner terms, and last terms), we can multiply the binomials on the right giving us:

In order to solve this equation, we need to have one side of the equation equal to .

Add to both sides.

Now, we can solve by using the quadratic formula, or by factoring. If we factor, we get:

Solving for each equation leaves us with the following answers:

and

In order to solve, we need to first cross-multiply.

Using FOIL (method of simplifying binomials by multiplying following the pattern of first terms, outer terms, inner terms, and last terms), we can multiply the binomials on the left giving us:

In order to continue solving this quadratic, we need to subtract from both sides so that the quadratic is equal to .

Now, we can solve using the quadratic formula:

Where the quadratic equation follows the pattern:

Therefore, we can use the terms in our quadratic equation and rewrite the equation as follows:

Since we can re-write as and , our answer becomes:

When solving two rational expressions that are set equal to each other Cross Multiply.

In this case we will multiply 4 by the the (3x-2), and also multiply 5 by the (2x+7)

Distribute on both sides to get:

Subtract 10x from both sides to get:

Add 8 to both sides to get:

Divide both sides by 2 to get the final solution:

Finally, double check for extraneaous solutions. Anytime you might get a zero in the bottom of a fraction, this is considered extraneaous because it is a mathematical impossibility to divide by zero.

In this case there are no x values in the denominators of the original problem, so there are no extraneaous solutions and the only solution will be:

We start by taking the original function, and replacing all the 's with . We end up with:

Then we can solve like normal:

If we noticed it, we also could have factored the numerator into:

The terms would have canceled leaving:

Solving that would have been much easier:

Either way, we would still get the exact same answer.

When we first look at this problem, we might be panicking because just plugging into the function makes us divide by , which we don't like. For now, let's forget about the denominator and focus on the numerator. If we look closely, we can see that we can factor the numerator:

Giving us:

As we can see, the will cancel from the numerator and the denominator, clearing up our "divide by 0" problem:

Now we can solve: