Rational Expressions & Equations — Challenge
Fractions with letters in them obey the fraction rules you already know — plus one forbidden value.
What these problems 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
Simplify x2x, where x is not zero.
- x
- x2
- 1
- 1x
- 0
Worked answer
x
Dividing powers of the same base subtracts the exponents: x2-1 = x1.
So the expression simplifies to x for every non-zero x.
The restriction matters: at x=0 the original expression has no value.
Problem 2
For what value of x is 1x-2undefined?
- 0
- 1
- 2
- -2
- It is never undefined
Worked answer
2
A fraction has no value when its denominator is zero.
x-2 = 0 when x = 2.
Every other value of x is fine, so the domain is all real numbers except 2.
Problem 3
Simplify x2-1x-1, where x is not 1.
- x-1
- x+1
- x
- x2
- 1
Worked answer
x+1
Factorise the numerator as a difference of squares: x2-1 = (x-1)(x+1).
The factor x-1 cancels with the denominator, leaving x+1.
The cancellation is only valid for x≠ 1, which is why that value is excluded.
Problem 4
Simplify 1x+1y.
- 1x+y
- x+yxy
- 2xy
- xyx+y
- x+y2
Worked answer
x+yxy
The common denominator is xy.
Rewrite each fraction: 1x = yxy and 1y = xxy.
Adding gives x+yxy. Adding denominators to get 1x+y is the classic error.
Problem 5
Simplify x2-4x2-2x.
- x-2x
- x+2x-2
- x+2x
- 2x
- x+2
Worked answer
x+2x
Factorise both parts: the top is (x-2)(x+2) and the bottom is x(x-2).
The common factor x-2 cancels, leaving x+2x.
Only common factors may cancel, never individual terms, so the x2 parts cannot be struck out.
Problem 6
Solve 1x=4.
- 4
- 14
- -4
- 12
- 0
Worked answer
14
Multiply both sides by x: 1 = 4x.
Divide by 4: x = 14.
Check: 11/4 = 4.
Problem 7
What is the vertical asymptote of y=1x-3?
- x=-3
- x=3
- y=3
- y=0
- x=0
Worked answer
x=3
A vertical asymptote occurs where the denominator is zero and the function blows up.
x-3 = 0 at x = 3.
The line y=0 is the horizontal asymptote, which is a different feature of the same graph.
Problem 8
As x becomes very large, what does 1x approach?
- 1
- 0
- It grows without bound
- -1
- It is undefined
Worked answer
0
Dividing 1 by a larger and larger number gives a smaller and smaller result: 110, 11000, 11000000.
The values get arbitrarily close to 0 without ever reaching it.
So the limit is 0, and y=0 is a horizontal asymptote of the graph.