Question 233·Medium·Equivalent Expressions
Which of the following is equivalent to the expression above for all values of for which both expressions are defined?
For rational-expression equivalence questions, first factor any polynomial denominators, then rewrite each term so all fractions share a common denominator. Combine the numerators carefully, paying close attention to distributing negative signs, and simplify the resulting numerator. Finally, factor the numerator if possible and compare your simplified form to the answer choices, checking that no illegal cancellations are made and that any domain restrictions (where denominators are zero) are consistent.
Hints
Start by factoring
Look at the in the denominator. Can you factor it as a difference of squares?
Get a common denominator
Once you factor , think about what denominator both fractions need to share so you can combine them.
Be careful with subtraction
When you rewrite the second fraction with the common denominator, you will subtract something like from . Make sure you distribute the subtraction across everything inside the parentheses.
Simplify and factor
After you combine the numerators, simplify the resulting expression and see if the numerator has a common factor you can factor out.
Desmos Guide
Enter the original expression
In Desmos, type f(x) = 5x/(x^2 - 9) - 2/(x - 3) to define the original function. Notice there are vertical asymptotes (gaps) at and where it is undefined.
Enter each answer choice as a separate function
Type gA(x) = 3(x + 2)/((x - 3)(x + 3)), gB(x) = 3(x - 2)/((x - 3)(x + 3)), gC(x) = 3(x - 2)/(x - 3), and gD(x) = 3(x - 2)/(x + 3).
Compare the graphs of these with the graph of .
Check which graph matches the original
The correct choice will have a graph that lies exactly on top of for all where both are defined (they can differ only at the vertical asymptotes). You can also graph f(x) - gChoice(x) for each choice; the correct one will give a difference of everywhere it is defined.
Step-by-step Explanation
Factor the quadratic denominator and note restrictions
First, factor the denominator in the first fraction:
So the original expression becomes
Also note that cannot be or , because those values make a denominator zero.
Rewrite with a common denominator
To combine the fractions, they must have the same denominator. The common denominator is .
The first fraction already has this denominator. For the second fraction, multiply the numerator and denominator by :
Now the expression is
Combine the fractions and simplify the numerator
With a common denominator, subtract the numerators and keep the denominator:
Distribute the in the numerator and simplify:
So the whole expression becomes
Factor the numerator and match the answer choice
Factor the numerator :
So the simplified expression is
which is equivalent to the original expression for all where both are defined (that is, and ). This matches answer choice B.