Rational Expressions & Equations — Practice
Fractions with letters in them obey the fraction rules you already know — plus one forbidden value.
What these 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.
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