Solving an equation that contains rational expressions is similar to solving any old equation that has fractions in it. In fact, if the variable only occurs in numerators, we actually are solving a equation with fractions in it. Our apologies to those of you who were psyched for something completely new and challenging. You'll need to get your kicks elsewhere.
Remember that, with fractions, the denominator tells us the size of the pieces, and the numerator tells us how many pieces we have. In order for two fractions with the same denominator to be equal, the numerators must also be the same.
Two pieces of size eight and three pieces of size eight will never be the same amount. For two pans of brownies to have the same amount of brownie goodness in them when the pieces are cut the same size, each pan must also have the same number of brownies. PS, in case you now absolutely must make a pan of brownies, we like this recipe.
Solve the equation .
Since the denominators are equal, the numerators must be equal, too. Ever since the Variable Rights Act, anyway. The only value of x that will work is 3.
Solve the equation .
There are two ways to do this.
Way 1: Eliminate the denominators. If you've got a bottle of Denominator-Off, that would work best. If not, have no fear. We can still make this happen.
If we multiply each side of the equation by 5, we find .
If we then multiply each side of the equation by 6, we get 18 = 5x, which means .
Way 2: Put the fractions over a common denominator so that we can compare the numerators, in the same way that it's easier to compare the size of two hot dogs when they're sitting on the same plate in front of you. If you've got one dog in your hand and the other at the opposite end of a football field, it'll be harder to tell the size difference. By the way, it seems to us you're misusing that football field.
The LCD of the two fractions is 5 × 6 = 30, so we can rewrite the left side of the equation like this:
And we can rewrite the right side of the equation as:
Now we solve the equation , which is equivalent to the original equation. Since the denominators are now the same, we need the numerators to be the same. We're in a "same" kind of mood right now.
We must have 18 = 5x, so divide both sides by 5 to finish up:
Both ways of solving the equation led us to the exact same answer. When asked to solve an equation involving rational expressions, you can use either method, whichever one floats your canoe:
Now it's time to take off the gloves. They were scrunching our fingers anyway. We'll allow the variables into the denominators this time as well. Sorry, denominators. Your club is exclusive no more. You need to keep up with the times.
When there are variables in denominators, the first thing to do is figure out which values of the variables make the denominators equal to zero. These values can't be solutions to the equation; we can't even evaluate both sides of the equation at these values. Unfortunately, we may need to factor to find the "bad" values of the variable. Then we'll give them an after-school detention and send a note home to their parents.
The value x = 2 can't possibly be a solution to the equation , since the left-hand side of the equation isn't even defined at x = 2. It's true; we checked dictionary.com and it wasn't there.
Similarly, we can factor the denominator on the right-hand side to see which values won't work over there:
x2 + 2x + 1 = (x + 1)(x + 1)
We see that x = -1 can't be a solution because the right-hand side of the equation isn't defined at that point.
After finding the "bad" values (naughty, naughty values), we can solve an equation with rational expressions using either of the two ways mentioned above. If you're having trouble deciding, flip a coin. If you're having trouble deciding which coin to use, roll a die. If you're still having trouble, we won't be able to help you. Those are the only ways we know to decide things.
Solve the equation .
First we check for "bad" values: x = 1 isn't allowed in this equation, since that would make more than one of our denominators zero. We already don't like it when one denominator equals zero. Can you imagine our wrath if two of them had that problem? You wouldn't like us when we're angry.
Now on to solving. We know you're raring to go. While we could add the expressions on the left-hand side of the equation and then put both sides of the equation over a common denominator, that sounds like too much work. For an easier way out, we'll stick with Way 1. To eliminate denominators, we multiply both sides of the equation by (1 – x). This gives us:
x – 1(1 – x) = 1
And that simplifies to x – 1 + x = 1, or 2x = 2.
This looks like x = 1 should be a solution to the equation. Alas, we found that x = 1 was a "bad" value. It can't be a solution after all, so the poor equation has no solutions. Maybe we can take up a collection for it. Sadface.
The value x = 1 in the example above was an extraneous solution, or a value that pretends to be a solution but can't actually be one because it's a "bad" value. Unlike a regular old "bad" value that's up to no good, an extraneous solution pretends to be your friend at the same time, which is even worse. Can you say "betrayal"?
Be careful: Check for extraneous solutions. Look for "bad" values before you solve a rational equation. After you solve the rational equation, only keep those solutions that aren't "bad" values. Keep them in a safe place, where the "bad" values can't get to them and turn them to the Dark Side.
Math is about translating English into funny symbols. Not laugh-out-loud funny, usually, but some of them should at least bring a wry smile to your face. What English statements translate into polynomials and rational expressions?
Polynomials often appear in problems where one quantity depends on another. In other words, the quantities are clingy.
Janna has finished weaving a blanket. She made the length of the blanket 1 foot greater than twice its width, because otherwise her toes get cold. If the area covered by the blanket is 28 square feet, how long is the blanket?
Whenever it makes sense to do so, draw a picture. You can draw a picture when it doesn't make sense to do so, but it's rare that a doodle of Mickey Mouse will help you solve a word problem, unless you're asked to find the circumference of his ears.
We need a variable somewhere. We'll use a variable for the width since the problem discusses the length in relation to the width.
If the blanket's width is x feet, its length must be 2x + 1 feet—twice the width plus 1. That means the area covered by this blanket is:
length of blanket × width of blanket = (2x + 1)(x) = 2x2 + x
Since we're told the area covered by the blanket is 28 square feet, we can set up an equation:
2x2 + x = 28
To solve this equation, we need to rearrange the terms so that we have a polynomial set equal to 0:
2x2 + x – 28 = 0
Then we factor:
(2x – 7 )(x + 4) = 0
The two solutions to this equation are x = 3.5 and x = -4. Do these solutions make sense? Welp, x = -4 sure doesn't make sense, because we can't have a blanket that's -4 feet wide. That won't cover even one of Janna's toes. The blanket must be 3.5 feet wide.
Wait, we hope you wrote that down in pencil. We need to make sure we're answering the right problem. We're supposed to give the length, not the width, of the blanket. (Slaps forehead with palm.) To find the length, we plug the width 3.5 in for x in the expression 2x + 1. The length of the blanket is:
2(3.5) + 1 = 8 feet
We can easily check our answers by multiplying the width and length of the blanket, and seeing if we do end up with 28.
3.5 × 8 = 28
Yup, we did it right.
Be careful: When solving word problems involving polynomials or rational expressions, make sure that you only keep those solutions that make sense in the context of the word problem. This often means throwing out one or more negative solutions. They're only bringing everybody down anyway.
There are a few English words and phrases that are frequently used to indicate rational expressions.
A ratio is a comparison between two quantities. We could say "the ratio of books to movies in this house is 3 to 1." Therefore, there are 3 books for every movie. Apparently, the person who was quoted as saying this lived in the 1940s.
A ratio can also be written using a colon like 3:1, or using fraction notation like .
That last bit with the fraction notation is how we find rational expressions. And you thought the stork brought them.
The ratio of girls to boys at Maria's school is 2:3, which could either be a good or a bad thing for Maria, depending on who you ask. If there are 22 girls at Maria's school, how many boys are there?
The ratio of girls to boys is . It's also , where x is the number of boys, so .
To solve this equation, we multiply both sides by 3 and then both sides by x:
2x = 66
x = 33
So there are 33 boys at Maria's school, and not one of them knows how to treat a lady.
That example was also an instance of a proportion, which is an equation that says two ratios are equal. It's "pro-equal portions," if you want to think of it that way. If you don't, that's cool, too. The equation is a proportion.
If Liana needs $5 to buy 6 pencils, how much money does she need to buy 7 pencils? Also, what in the world is she doing spending so much on pencils? Were these famous pencils?
We can set this up as a proportion:
...where x is the amount of money Liana needs, assuming she hasn't come to her senses and picked up a $1.49 12-pack from Office Depot.
Multiplying both sides by 7 gives .
Since , Liana would need $5.84 to buy 7 pencils.
Another idea that appears around rational expressions is the idea of proportionality. We say y is directly proportional to x if:
y = (some constant)(x)
The constant is called the constant of proportionality. Where did they come up with that one?
If y = 5x, then y is directly proportional to x and the constant of proportionality is 5.
If y is directly proportional to x, it means two things. Or at least two things, but these are our favorites. We have a list somewhere with a ranking of our top four hundred, if you'd like to see it.
Assume that Tina walks at a constant rate. In other words, she doesn't stop along the way to pick up any pennies or caterpillars. The distance she travels is directly proportional to the time she walks, where the proportionality constant is the rate at which she walks. If Tina travels 3.5 miles over an hour and fifteen minutes, how fast did she walk? Follow-up question: how much money could she have had if she had picked up all those pennies?
There are so many words here, including many that are utterly silly, they're obscuring what's going on. Let's translate into math a little at a time. Someone summon the interpreter.
We're told "the distance she travels is directly proportional to the time she walks," which means:
distance = constant × time
Since we're told the proportionality constant is her rate, we use this equation, which may already be familiar to you:
distance = rate × time
Bells should be going off in your head, and not the ones you usually hear whenever you look into a bright light. You should see a doctor about that, by the way. Tina's distance was miles and she walked for hours, or an hour and fifteen minutes.
We can solve this by eliminating the denominators. Multiply both sides by 4.
14 = (rate) × 5
So her rate was miles per hour. Hey Tina, where's the fire?
For the sake of completeness, there's also something called inverse proportionality. Not to be confused with Converse proportionality, which makes sure that both of your shoes are the same size. We say y is inversely proportional to x if:
If , then y is inversely proportional to x with a constant of 7.
If y is inversely proportional to x, then: