How to Solve a Rational Equation

Video Lesson on How to Solve a Rational Equation

This lesson video Not available at this time available video coming soon

How to Solve a Rational Equation

A rational equation is one that contains fractions. Yes, we will be finding a common denominator that has 'x's. But no worries! Together we will use a process that will help us solve rational equations every time!

Rational Equations

A rational equation is an equation that contains fractions with xs in the numerator, denominator or both. Here is an example of a rational equation: (4 / (x + 1)) - (3 / (x - 1)) = -2 / (x^2 - 1).

Let's think back for a moment about solving an equation with a fraction. 1/3 x = 8. We think of the 3 in the denominator as being a prisoner, and we want to release it. To set the 3 free, we multiply both sides of the equation by 3. Think of it as 3 letting both sides of the equation know he's leaving. 3 (1/3 x) = 8 (3).

This process freed our denominator and got rid of the fraction - x = 24. It is also the process we use to solve rational equations with one extra step. In rational equations, sometimes our solution may look good, but they carry a virus; that is, they won't work in our equation. These are called extraneous solutions. The steps to solve a rational equation are:

  1. Find the common denominator.
  2. Multiply everything by the common denominator.
  3. Simplify.
  4. Check the answer(s) to make sure there isn't an extraneous solution.
  5. Let's solve a couple together.

    Example #1

    Example number one: solve. Remember to check for extraneous solutions. (3 / (x + 3)) + (4 / (x - 2)) = 2 / (x + 3).

    Our first step is to figure out the terms that need to be released from the denominators. I look at 3 / (x + 3). I write down (x + 3) as one of my common denominators. I look at 4 / (x - 2). I write down (x - 2) as another part of my common denominator. I look at 2 / (x + 3). Since I already have (x + 3) written in my denominator, I don't need to duplicate it.

    Next, we multiply everything by our common denominator - (x+3)(x-2). This is how that will look: ((3(x + 3)(x - 2)) / (x + 3)) + ((4(x + 3)(x - 2)) / (x - 2)) = (2(x + 3)(x - 2)) / (x + 3))

    It isn't easy for the denominators to be released; there is a battle, and like terms in the numerator and denominator get canceled (or slashed). Slash (or cancel) all of the (x + 3)s and (x - 2)s in the denominator and numerator. Our new equation looks like: 3(x - 2) + 4(x + 3) = 2(x - 2).

    35 -14390410

    In example #1, the first step is finding the common denominator.

    Distribute to simplify: (3x - 6) + (4x + 12) = 2x - 4. Collect like terms and solve. 3x + 4x = 7x, -6 + 12 = 6. We end up with 7x + 6 = 2x - 4.

    Subtract 2x from both sides: 7x - 2x = 5x. Subtracting from the other side just cancels out the 2x, and we get 5x + 6 = -4. Subtract 6 from both sides: -4 - 6 = -10. Again, subtracting 6 will cancel out the +6, so we end up with 5x = - 10. Divide by 5 on both sides, and we cancel out the 5 and give us x = - 2. It turns out x = - 2.

    The reason we check our answers is that sometimes we get a virus, or, in math terms, extraneous solutions. To check, I replace all the xs with -2: (3 / (-2 + 3)) + (4 / (-2 - 2)) = (2 / (-2 + 3)). Let's simplify: (3 / 1) + (4 / -4) = (2 / 1). Since 3 + -1 = 2 is true, x = - 2 is the solution!

    Example #2

    Example number two: solve. Remember to check for extraneous solutions. (4 / (x + 1)) - (3 / (x - 1)) = -2 / (x^2 - 1).

    First we need to release our denominators. To release our denominators, we write down every denominator we see. I have found the easiest way to do this is to first factor, if needed, then list the factors. x^2 - 1 = (x + 1)(x- 1).

    Our new equation looks like this: (4 / (x + 1)) - (3 / (x - 1)) = -2 / (x + 1)(x - 1).

    I look at 4 / (x + 1). I write down (x + 1) as one of my common denominators. I look at 3 / (x - 1). I write down (x - 1) as another part of my common denominator. I look at -2 / (x + 1)(x - 1). Since I already have those written in my denominator, I don't need to duplicate them. So my common denominator turns out to be (x + 1)(x - 1).

    Kathryn, why aren't we using the factors of x^2 - 1? Great question! We already have (x + 1) and (x - 1) being released. We don't need to do it twice.

    Now we multiply each part of the equation by the common denominator - (x + 1)(x - 1). Think of this as the key to the prison: (4 (x + 1)(x -1) / (x + 1)) - (3 (x + 1) (x - 1) / (x - 1)) = -2 (x + 1)(x - 1) / (x + 1)(x - 1).

    It isn't easy for the denominators to be released; there is a battle, and like terms get canceled (or slashed)! Slash (or cancel) all of the (x + 1)s and (x - 1)s in the denominator and numerator. This leaves us with 4(x - 1) - 3 (x + 1) = -2.

    36 -17193958

    Like terms are cancelled out or slashed in the second example.

    Now we need to solve for x. Distribute 4 into (x - 1) and -3 into (x + 1). (4x - 4) - (3x - 3) = -2. Collect like terms: x - 7 = - 2. Add 7 to both sides of the equal sign: x = 5.

    It looks like our answer is 5, but we need to double-check. I replace all the xs with 5 and simplify. It turns out 5 works, and it is the solution to our equation. And so our solution checks!

    Lesson Summary

    The steps to solving a rational equation are:

    1. Find the common denominator.
    2. Multiply everything by the common denominator.
    3. Simplify.
    4. Check the answer(s) to make sure there isn't an extraneous solution.

    Next Topics

Analytical Reasoning with Explained Questions
All in this Category