Rational Expressions And Equation Test

Rational Expressions and Equation Test Study Guide 3/4 3/7 3/11 3/13 3/18 3/20 4/1 4/3 4/4 4/11

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• Simplify rational expressions • Identify the domain (restrictions on the variable) • Add, subtract, multiply and divide rational expressions • Solve equations involving rational expressions • Solve work, distance and mixture problems • Factor quadratic expressions and solve quadratic equations • Distinguish between equations and expressions

Work Problems * 1. It takes Laura 15 minutes to pick apples from the tree. Rachel can do it in 25 minutes. How long will it take them working together? 2. It takes Robert 1 hour to milk all of the cows and it takes Maggie 1.5 hours. How long will it take them to do the job together? 3. Kabir can paint a kitchen in 20 hours and his assistant, Malik, can do it in 30 hours. How long will it take them together? 4. Malik can load his truck in 24 minutes. When his brother helps him, the job is done in 15 minutes. How long would it take his brother working alone? ** 5. One printing machine works twice as fast as another. When both machines are used they print a magazine in 3 hours. How many hours would each machine require to do the job alone? 6. Maddy can do a job in 30 minutes, Julia can do it in 40 minutes and it takes Lauren an hour. How long will it take them if they work together? 7. It takes my father 3 hours to weed the backyard. I can do the job in 5 hours. If we weed for 1 hour before I go to school, how much longer will it take my father? 8. One pump can fill a tank in 3 hours and another pump takes 5 hours. When the tank is empty both pumps are turned on for 30 minutes and then the faster pump was turned off. How much longer will it take to fill the tank using the slower pump? 9. Jimmy can do a job in 12 days. After working for 4 days Jesse joins him and the job is done after two more days. How long would it take Jesse working by himself? 10. Aly and Ari are stuffing envelopes for Joy. Aly can stuff and address one every 30 seconds. Ari can stuff and address 1 every 40 seconds. How long will it take them to stuff and address 140 envelopes?

11. If 5 students can do 280 math problems in 2 days. How many students are needed to complete 2100 math problems in 3 days? 12. Paul can weed the garden in 4 hours. His wife Cheryl takes the same amount of time. After working 1 hour, their son Matthew came home and the job was completed in 30 minutes. How long would it take Matthew working by himself? 13. If three pipes are opened, they fill an empty swimming pool in three hours. The largest pipe alone takes one-third the time that the smallest pipe takes and one half the time the other pipe takes. How long would it take each pipe working by itself? *** 14. Sruti, Katie and Christine are painting identical murals in four locations around Terman. Sruti and Katie paint the first mural in 6 hours. Sruti and Christine spend 8 hours working together to paint the second mural. Katie and Christine paint the third mural in 9 hours. All three work together on the fourth mural. How many hours will it take them? How long would it take the quickest of the three working alone?

Rational Expressions: Finding the Domain A "rational expression" is a polynomial fraction, and anything you could do with regular fractions you can do with rational expressions. However, since there are variables in rational expressions, there are some additional considerations. When you dealt with fractions, you knew that the fraction could have any whole numbers for the numerator and denominator, as long as you didn't try to divide by zero. When dealing with rational expressions, you will often need to evaluate the expression, and it can be useful to know which values would cause division by zero, so you can avoid these x-values. So probably the first thing you'll do with rational expressions is find their domains. •

Find the domain of

3

/x.

The domain is all values that x is allowed to be. Since I can't divide by zero (division by zero isn't allowed), I need to find all values of x that would cause division by zero. Then the domain will be all other x-values. And when is the denominator equal to zero? When x = 0. Then the domain is "all x not equal to zero" (x cannot equal 0) •

Determine the domain of

x

/3.

The domain doesn't care what is in the numerator of a rational expression. The domain is only influenced by the zeroes of the denominator. Will "3" ever equal zero? Of course not. Since the denominator will never equal zero, no matter what value x is, then there are no forbidden values, and x can be anything. So the domain is "all x". •

Give the domain of the following expression:

To find the domain, I'll ignore the "x + 2" in the numerator (since the numerator does not cause division by zero) and instead I'll look at the denominator. I'll set the denominator equal to zero, and solve. The x-values in the solution will be the x-values which would cause division by zero. The domain will then be all other x-values.

x2 + 2x – 15 = 0 (x + 5)(x – 3) = 0 x = –5, x = 3 By factoring the quadratic, I can find the zeroes of the denominator. The domain will then be all other x-values:

•

Find the domain of the following expression:

To find the domain, I'll solve for the zeroes of the denominator:

x2 + 4 = 0 x2 = –4 This has no solution, so the denominator is never zero. Then the domain is "all x".

Rational Expressions: Simplifying Thinking back to when you were dealing with whole-number fractions, one of the first things you did was simplify them: You "cancelled off" factors which were in common between the numerator and denominator, because dividing a number by itself gives you just "1", and you can ignore factors of "1". So, using the same reasoning and methods, let's simplify some rationals. •

Simplify the following expression:

To simplify a fraction, you cancel off any common factors. Considering the factors in this fraction, I get:

Then the simplified form is:

•

Simplify the following rational expression:

The common factor here is "x + 3", so I'll cancel that off and get:

Then the simplified form is:

The common temptation at this point is to try to continue on by cancelling off the 2 with the 4. But you cannot do this. Whenever you have terms added together, there are understood parentheses around them, like this:

You can only cancel off factors (that is, entire expressions contained within parentheses), not terms (that is, not just part of the contents of a pair of parentheses). To go inside the parentheses and try to cancel off part of the contents is like ripping off arms and legs of the poor little polynomial trapped inside. It'll be bleeding and oozing and flopping around on the floor, whimpering while looking at you sadly with big brown eyes... Well, okay; maybe not. But trying to cancel off only a portion of a factor would be like trying to do this:

Is 66/63 equal to 2? Of course not. So if the above "cancellation" is illegitimate, then so also is this one:

...and it's illegitimate for exactly the same reason as the previous one was. While it isn't quite so obvious that you're doing something wrong in the second case with the variables, these two "cancellations" are not allowed because you're reaching inside the factors (the 66 and 63 above, and the x + 4 and x + 2 here) and ripping off parts of them, rather than cancelling off an entire factor. Always remember: You can only cancel factors, not terms!

Note: When I went from the original expression:

...to the simplified form:

...I removed a "division by zero" problem. That is, in the original fraction, I could not have x = –3, because this would have caused division by zero. But in the reduced fraction, x was allowed to be –3. If the two expressions have different domains, can they really be equal? Not exactly. Depended upon the text you're using, this technicality with the domain may be ignored or glossed over, or else you may be required to make note of the domain. That is, many (most?) books will accept the answer:

...but some will require the simplified form to have the same domain, so the answer would be:

Depending on your book and instructor, you may not need the "as long as x isn't equal to –3" part. If you're not sure which answer your instructor is expecting, ask now (before the test).

Rational Expressions: More Simplifying •

Simplify the following rational expression:

Many students will try to do something like the following:

Is this legitimate? Can the student really do this? (Think "bleeding, oozing...") You cannot cancel term-by-term, because you can only cancel factors! So the first thing I have to do (if I'm going to do the simplification correctly) is factor the numerator and denominator:

Since there is a common factor, I can reduce:

Can I reduce any further? For instance, can I cancel off the x's? (whimpering, bleeding...) Can I cancel a 2 out of the 4 and the 6? (oozing, flopping...) No! This is as simplified as it's going to get, because there are no remaining common factors. Then the answer is:

Depending on your text, you might not need that "for x not equal to –5/2 part". However, since I cancelled off a "2x + 5" factor, this removed a division-by-zero problem, because 2x + 5 = 0 for x = –5/2. •

Simplify the following:

These factors are almost the same, but not quite, so they can't be cancelled — yet. If the fraction had been:

...(that is, plusses instead of minuses), I could have rearranged the terms as:

...and cancelled to get "1", since order doesn't matter in addition. But order does matter in subtraction, so I can't just flip the subtraction to get matching factors. However, take a look at this:

5–3=2 3 – 5 = –2 Do you see? If I reverse a subtraction, I get the same number, but with the opposite sign. That is, if I flip a subtraction, it kicks a "minus" sign out front. This means that I can reverse the subtraction above, as long as I remember to switch the sign out front. That is:

Now I can cancel:

Remember: If "nothing" is left, a "1" is left, so:

(Depending on the text you're using, you may or may not need the "as long as x does not equal 2" part.) You should keep this "flip a subtraction and kick a 'minus' out front" trick in mind. Depending on the text you're using, you may see a lot of this. •

Simplify the following expression:

To simplify this, I first need to factor. Then I can cancel off any common factors.

Then the answer is:

(You might not need the "for all x not equal to –3" part.) •

Simplify the following expression:

To simplify this, I need to factor; I'll also need to flip the subtraction in the denominator, so I'll need to remember to change the sign.

Then the answer is:

(You might not need the "for all x not equal to 6" part.) As you have probably noticed by now, simplifying rational expressions involves a lot of factoring quadratics. If you're feeling at all rusty on this topic, review now.

Multiplying Rational Expressions With regular fractions, multiplying and dividing is fairly simple, and is much easier than adding and subtracting. It is the same with rational expressions (polynomial fractions). The only major problem students have with multiplying and dividing rationals is illegitimate cancelling, where they try to cancel terms instead of factors, so I'll be making a big deal about that as we go along. Remember how you multiply regular fractions: You multiply across the top and bottom, and cancel off any duplicate factors. For instance:

You always need to simplify, where possible:

While the above is perfectly valid, it is generally simplest to cancel first and then multiply, since you'll be dealing with smaller numbers that way:

This process (cancel first, then multiply) works with rationals, too. •

Simplify the following expression:

Simplify by cancelling off duplicate factors:

Then the answer is:

Why did I add "for x not equal to 0"? Because the domain of the original function did not include x = 0 (since this would have caused division by zero). For the two expressions to be technically

equal, their domains have to be the same. Since 3x/2 has no problem at x = 0, I have to explicitly state this exclusion. Your text might not make this distinction. If you're not sure if your teacher cares about this technicality, make sure you find out before the test. •

Multiply and simplify the following:

Many students find it helpful to convert the "15" into a fraction. This can make it a little more obvious what cancels with what.

Can you cancel off the 2 into the 20? No! When you have a fraction like this, there are understood parentheses around any sums of terms, like this:

You can only cancel off factors (the entire contents of a set of parentheses), not terms (one of the addends inside a set of parentheses). Going inside the parentheses and hacking x's and y's and arms and legs off the poor polynomial doesn't simplify anything; it just leaves the little polynomial lying there on the floor, quivering and bleeding and oozing and whimpering... Okay, maybe not; but you get the point: Never reach inside the parentheses and hack off part of the contents. Either you cancel off the entire contents with a matching factor from the other side of the fraction line, or you don't cancel anything at all. (I told you I'd be making a big deal of this!) The only thing that factors out of the 20x + 25 is a 5, and that doesn't cancel off with the 2 underneath, so, for this rational, there is no further reduction to be done. Then the final answer is:

•

Multiply and simplify the following expression:

Some students, when faced with this problem, will do something like this:

Can you really "cancel" like this? (Think "bleeding"...) Is this even vaguely legitimate? (Oozing...) Has this student done anything at all correctly? (Flopping, whimpering...) No, no, and no! You can not cancel terms; you can only cancel factors. So my first step has to be to factor. Once I've factored everything, I can cancel off any factor that is mirrored on either side of the fraction line. The legitimate simplification looks like this:

Can I now cancel off the 2's? (Tears are welling up in the polynomial's eyes...) Can I cancel off the x's with the x2? (Now the polynomial is starting to cry...) Hmm??? No! The x's are only part of their factors; they are not stand-alone factors, so they can't cancel off with anything. Then the answer, including the trouble-spots that I removed from the domain when I cancelled the common factors, is:

The "x not equal to 0, –1 or –3" came from the factors that I cancelled off; your book may not require this information. Note: For reasons which will become clear when you are adding rational expressions, it is customary to leave the denominator factored, as shown above. At this stage, your book may or may not want the numerator factored. You should recognize, in any case, that "(2x – 5)(x + 2)2"

is the same thing as "2x3 instructor expects it.

+ 3x2 – 12x – 20", and convert the form of your answer if your book or

Dividing Rational Expressions For dividing rational expressions, just remember that, when dividing by a fraction, you flip-nmultiply. For instance: •

Perform the indicated operation:

To simplify this division, I convert it to multiplication by flipping what I'm dividing by, and then I simplify as usual:

Can the 2's cancel off from the 20's? No! This is as simplified as the fraction gets. Division works the same way with rational expressions. •

Perform the indicated operation:

To simplify this, first I flip-n-multiply. Then, to simplify the multiplication, I factor and cancel. It looks like this:

Then the answer is:

The problems you'll be given won't usually simplify that nicely, though. This example is more typical:

•

Simplify the following expression:

First, I need to flip the fraction I'm dividing by, converting to multiplication. Then I'll factor, and see if anything cancels.

(Can you cancel off the 6's? or the x's? No! The above is as simplified as this gets!) Then the final answer is:

For reasons which will become clear when adding and subtracting rationals, the numerator is usually multiplied out, while the denominator is usually left in factored form.

Make sure you know how to factor quadratics and cubics, because, as you have seen, it is required for many of the problems you'll be doing. Also, make sure you are careful to cancel only factors, not terms. If you can keep that straight, then you'll probably do fine.

Adding and Subtracting Rational Expressions: More Examples •

Simplify the following:

To find the common denominator, first I'll have to factor the quadratic denominator:

x2 – 5x – 6 = (x – 6)(x + 1) Fortunately for me, the quadratic denominator didn't introduce any new factors to the problem, so the common denominator will be (x – 6)(x + 1).

Since I was able to cancel out the x + Then the final answer is:

1 factor, I eliminated a zero from the denominator.

You might not need the "for x not equal to –1" part of the solution. •

Simplify the following:

First I'll factor the quadratic denominator:

x2 + 3x – 10 = (x + 5)(x – 2) Note that these factors almost match the other denominators, but the second fraction's denominator is backwards. How can I fix that? By remembering the following:

5–3=2 3 – 5 = –2 The point of this is that, when I reversed the subtraction, I got the same answer except for the sign. So I can reverse the subtraction in the second fraction's denominator, as long as I remember to reverse the sign. This is what that looks like:

I factored the numerator, but nothing cancels out. As you can see, I had to factor a denominator, multiply two of the fractions to get a common denominator, multiply those two fractions' numerators, add, simplify, and then factor again. You should expect to see some problems that are at least this involved. They're not as much complicated as they are long and annoying. Work them out step-by-step as I did above, and you'll get the right answers fairly regularly. In this case, the answer is:

When you're adding and subtracting rationals, don't try to do a lot of steps in your head, or skip steps or do half-steps (like leaving out the denominators in your calculations), or you'll pretty much guarantee yourself the wrong answer. Take the time to do every step, and practice them on the homework, so you have a good chance of getting these right on the test.

Equation • • •

An equation is a SENTENCE. One solves an equation. An equation HAS a relation symbol. Ex.

1. Ten is five less than a number. 10 = x - 5 2. A number is less than five. x<5 Equation, Not Expression Computation Errors

Expression, Not Equation Computation Errors

• • •

Expression An expression is a PHRASE, a sentence fragment. One simplifies an expression. An expression HAS NO relation symbol. Ex.

3. a number less than five x 4. five less than a number x-5