دوره GRE Test- Practice & Study Guide ، فصل 21 : GRE Quantitative Reasoning- Rational Equations and Expressions

درباره‌ی این فصل:

Review elements of rational equations and expressions to prepare for the Graduate Record Exam (GRE). Video lessons and quizzes incorporate these elements of the quantitative reasoning section in a simple and clear manner.

Rational Equations

A rational equation is an equation that contains fractions with x s 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.

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).

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

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

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

Like terms are cancelled out or slashed in the second example. Cancelling Like Terms Rational Equation

Now we need to solve for x . Distribute 4 into ( x - 1) and -3 into ( x + 1). (4 x - 4) - (3 x - 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 x s 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.

این مجموعه تلوزیونی شامل 6 فصل زیر است:

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!

Mario and Bill own a local carwash and have several complex tasks that they must use rational equations to solve for an answer. Enjoy learning how they solve these equations to help them with some of their day-to-day tasks.

Multiplying and dividing rational polynomial expressions is exactly like multiplying and dividing fractions. Like fractions, we will reduce. With polynomial expressions we use factoring and canceling. I also give you a little mnemonic to help you remember when you need a common denominator and when you don't.

Adding and subtracting rational expressions brings everything you learned about fractions into the world of algebra. We will mix common denominators with factoring and FOILing.

Adding and subtracting rational expressions can feel daunting, especially when trying to find a common denominator. Let me show you the process I like to use. I think it will make adding and subtracting rational expressions more enjoyable!

Let's continue looking at multiplying and dividing rational polynomials. In this lesson, we will look at a couple longer problems, while giving you some practice multiplying and dividing.

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