Rational Expressions & Equations — Print
Fractions with letters in them obey the fraction rules you already know — plus one forbidden value.
What the practice questions cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Reciprocal — The flip of a fraction. The reciprocal of 35 is 53.
- LCD — Least common denominator — the smallest number both denominators divide into.
- Factor — A number that divides evenly into another.
- Asymptote — A line the graph approaches but never reaches — here, the x-axis.
- Leading coefficient — The number in front of the highest power. Its sign decides which way the right end goes.
- Domain restriction — A square root needs its inside to be ≥ 0, which limits the inputs allowed.
- Vertical asymptote — A vertical line the graph approaches but never reaches, at an excluded value that does not cancel.
- Horizontal asymptote — The height the graph settles towards far out to the left and right.
One tap fills the tub in 3 hours, the other in 6.
Turn both on. A tempting answer is the average, four and a half hours, but that cannot be right.
Two taps must be faster than either one alone, so the answer has to be under 3 hours.
What adds is not the times, it is the rates: 13 of a tub per hour plus 16 gives 12.
Half a tub per hour means two hours. Adding rates means adding fractions with letters in them — which is this whole unit.
Simplify by factoring, never by crossing out terms
A rational expression simplifies only when top and bottom share a whole factor.
x2-9x+3 factors to (x-3)(x+3)x+3, and the (x+3) cancels to leave x - 3.
But x = -3 is still excluded, because the original expression was undefined there.
Factors cancel, terms do not
x+2x is not 2. The x on top is added, not multiplied, so nothing cancels. Factor first, then look.
Multiplying and dividing
Multiplying is the easy one: factor everything, cancel across, and multiply what is left.
Dividing is multiplying by the reciprocal, exactly as with numbers. Flip the second fraction and carry on.
Cancel before you multiply out — expanding first creates large polynomials you then have to factor again.
Cancel early
Factoring first turns a page of algebra into two lines. The temptation to expand is almost always the longer road.
Adding needs a common denominator
As with numbers, you cannot add until the bottoms match. Factor each denominator to find the least common denominator.
For 1x + 1x+2 the LCD is x(x+2), so the fractions become x+2x(x+2) and xx(x+2).
Add the tops, leave the bottom, then factor the result to see whether anything cancels.
Multiply the tops too
Rewriting 1x over x(x+2) means multiplying top and bottom by (x+2). Changing only the denominator changes the value.
Solving: clear the fractions
An equation is different from an expression — here you may multiply every term by the LCD and be rid of the fractions entirely.
That turns a rational equation into a polynomial one, which you already know how to solve.
Then check each answer against the excluded values, because multiplying by the LCD can introduce a root that was forbidden all along.
Expression or equation?
You may clear denominators only when there is an equals sign. In an expression, multiplying by the LCD changes what you were given.
Asymptotes
An excluded value that does not cancel gives a vertical asymptote — the graph shoots off beside that line.
For the far left and right, compare the degrees: equal degrees give a horizontal asymptote at the ratio of the leading coefficients.
If the bottom has the higher degree, the graph settles on y = 0, because the denominator wins the race.
A hole, not an asymptote
If every copy of the denominator factor cancels and the reduced expression is defined at that value, the graph has a hole. If a denominator factor remains, there is still a vertical asymptote.
Worked examples
Simplifying
Simplify x2 - 4x2 + 5x + 6 and state the excluded values.
- Factor both: (x-2)(x+2)(x+2)(x+3). Nothing can cancel until both are in factors.
- Cancel (x+2) to get x-2x+3. A whole factor, top and bottom.
- Excluded: x = -2 and x = -3. Both came from the original denominator, and -2 stays excluded even though it cancelled.
x-2x+3, x ≠ -2, -3
Adding
Simplify 3x + 2x+1.
- The LCD is x(x+1). The denominators share nothing, so the LCD is their product.
- Rewrite: 3(x+1)x(x+1) + 2xx(x+1). Multiply each top by whatever its bottom was missing.
- Add the tops: 3x + 3 + 2xx(x+1) = 5x+3x(x+1). The top does not factor, so this is finished.
5x+3x(x+1)
The two taps
One tap fills a tub in 3 hours, another in 6. How long together?
- Rates add: 13 + 16 of a tub per hour. Times do not add; rates do.
- Common denominator 6: 26 + 16 = 36 = 12. Half a tub per hour.
- Half a tub per hour means one tub in 2 hours. Under 3, as it had to be.
2 hours
Words to know
- Rational expression
- A fraction whose top and bottom are polynomials.
- Excluded value
- An input that makes the denominator zero. It can never be in the domain.
- Vertical asymptote
- A vertical line the graph approaches but never reaches, at an excluded value that does not cancel.
- Horizontal asymptote
- The height the graph settles towards far out to the left and right.
- Least common denominator
- The smallest expression every denominator divides into.
Mistakes to watch for
Cancelling terms instead of factors. In x+2x nothing cancels, because the x on top is added rather than multiplied. Only a factor shared by the whole top and the whole bottom may go.
Losing an excluded value after it cancels. x2-9x+3 simplifies to x-3, but x = -3 is still forbidden — the original expression was undefined there, and simplifying does not undo that.
Adding the denominators. 1x + 1x+2 is not 22x+2. Find a common denominator and add only the tops, exactly as with numbers.
Clearing denominators in an expression. Multiplying by the LCD is allowed only across an equals sign. In an expression it changes the value you were asked to simplify.
Averaging the times in a work problem. Two taps taking 3 and 6 hours do not average to 4.5; together they must beat 3. Add the rates, not the times.
Work problems like this appear in a Chinese text from around 100 BC, with the same tubs and the same trap.
The forbidden values, where the bottom is zero, are what give rational graphs their vertical asymptotes — the lines the curve races towards and never touches.
Engineers call the ratio of two polynomials a transfer function, and use it to describe everything from a loudspeaker to a suspension bridge.
Problem 1
What value must be excluded from 5x-3?
Worked answer
3
x-3=0 at x=3
Problem 2
What value must be excluded from x2x+8?
Worked answer
-4
2x+8=0
x=-4
Problem 3
Simplify 6x23x.
Worked answer
2x
6 ÷ 3 = 2
x2 ÷ x = x
Keep the original domain restriction: x≠0.
Problem 4
Simplify x2-9x+3.
Worked answer
x-3
(x-3)(x+3)x+3
Cancel the shared factor
Keep the original domain restriction: x≠-3.
Problem 5
Simplify 2x · x5.
Worked answer
2/5
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Keep the original domain restriction: x≠0.
Problem 6
Simplify x2-4x2+4x+4.
Worked answer
(x-2)/(x+2)
(x-2)(x+2)(x+2)2
Cancel one (x+2)
Keep the original domain restriction: x≠-2.
Problem 7
Simplify 2x+3x.
Worked answer
5/x
2+3x
Problem 8
Simplify 1x+12x.
Worked answer
3/(2x)
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Problem 9
Simplify x3 ÷ x6.
Worked answer
2
x3 · 6x
=63=2
Keep the original domain restriction: x≠0.
Problem 10
Solve 6x=3.
Worked answer
2
6=3x
x=2
Problem 11
Solve x4+x6=5.
Worked answer
12
3x+2x=60
5x=60
x=12
Problem 12
One pipe fills a tank in 4 hours, another in 12. How many hours together?
Worked answer
3
14+112=312+112=412=13
A third of a tank per hour means 3 hours
Problem 13
One worker paints a room in 6 hours, another in 3. How many hours together?
Worked answer
2
16+13=16+26=36=12
Half a room per hour means 2 hours
Problem 14
What is the vertical asymptote of y=x+1x-5?
- x=5
- x=-1
- y=5
- y=1
Worked answer
x=5
x-5=0 at x=5
The top is not zero there, so it is an asymptote
Problem 15
What is the horizontal asymptote of y=3x+2x-4?
- y=3
- y=0
- y=4
- There is none
Worked answer
y=3
Equal degrees, so take the ratio of the leading coefficients
31=3
Problem 16
Solve 1x+13=12.
Worked answer
6
6+2x=3x
x=6