A reliable method
Factor numerator and denominator, cancel shared factors rather than individual terms, and carry the original restrictions into the simplified form.
The common trap
The expressions (x² − a²)/(x − a) and x + a agree where the first is defined, but the first still excludes x = a. A simplified formula does not fill a hole automatically.
Check and transfer
To combine rational expressions, create a common denominator. Check numerical values at allowed inputs, and never substitute an excluded input simply because a simplified form accepts it.
Worked comparison: identify the structure
For R(x)=(x²−16)/(x−4), cancellation gives x+4 only when x≠4. At x=5 both forms give 9; at x=4 the original remains undefined. The canceled factor creates a missing input, not permission to divide by zero.
Transfer and explain
Combine 2/x+1/(3x) as 6/(3x)+1/(3x)=7/(3x). Adding denominators gives 3/(4x), which disagrees at x=1. Preserve restrictions when solving related equations.
The expression (x + 2)/(3x − 9) is undefined at what value of x?
- A rational expression is undefined when its denominator is zero.
- Solve the denominator equation; the numerator does not determine this exclusion.
- The requested value is 3. Substitute or reverse the operation to check this result.
Compare the other answer choices
Forget to divide the denominator constant by its x coefficient.
Reverse the sign when solving the denominator equation.
Set the numerator equal to zero instead of the denominator.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
CLEP College Algebra exam overviewSiyavula Grade 10 foundation reference