Question 210·Hard·Equivalent Expressions

Once you factor $r^2-4$, think about what denominator all three fractions can share. Rewrite $\dfrac{5r}{r-2}$ and $\dfrac{4r}{r+2}$ so they both have the denominator $r^2-4$.

After all three fractions have the same denominator, combine them into a single fraction. Then carefully simplify the numerator—combine like terms and see if you can factor it.

When you have a single simplified fraction, compare its numerator and denominator with each answer choice. Pay close attention to signs (plus vs. minus) and whether the denominator is $r^2-4$ or $r^2+4$.

In one line, type f(r) = 5r/(r-2) - 4r/(r+2) + 8r(r-2)/(r^2-4). Use the table feature to create an $r$-column and evaluate $f(r)$ at several values such as $r=-3, -1, 0, 1, 3$ (avoid $r=\pm 2$ since the expression is undefined there).

On new lines, enter gA(r) = r(9r-2)/(r^2-4), gB(r) = r(9r+2)/(r^2-4), gC(r) = r(9r+2)/(r^2+4), and gD(r) = r(9r-2)/(r^2+4). Turn on the table for each so you can see their values at the same $r$-values you used for $f(r)$.

For each chosen $r$-value, compare $f(r)$ with $gA(r)$, $gB(r)$, $gC(r)$, and $gD(r)$. The function whose values match $f(r)$ for all tested $r$-values corresponds to the equivalent expression. You can also look at the graphs: the correct choice will produce a graph that coincides with the graph of $f(r)$ (except at the points where the expressions are undefined).

First, notice that

So the common denominator for all three fractions is $r^2-4$.

Rewrite each term so it has denominator $r^2-4$:

• $\dfrac{5r}{r-2}$: multiply top and bottom by $(r+2)$.
• $\dfrac{4r}{r+2}$: multiply top and bottom by $(r-2)$.
• $\dfrac{8r(r-2)}{r^2-4}$ already has denominator $r^2-4$.

Carry out the rewriting:

• $\dfrac{5r}{r-2} = \dfrac{5r(r+2)}{(r-2)(r+2)} = \dfrac{5r(r+2)}{r^2-4}$
• $\dfrac{4r}{r+2} = \dfrac{4r(r-2)}{(r+2)(r-2)} = \dfrac{4r(r-2)}{r^2-4}$

So the whole expression becomes

Now that the denominators match, you can combine the numerators.

Combine the three fractions into one by adding/subtracting the numerators over the common denominator:

Group the terms with $(r-2)$:

So the numerator becomes

Now simplify this numerator.

Expand each product in the numerator:

• $5r(r+2) = 5r^2 + 10r$
• $4r(r-2) = 4r^2 - 8r$

Add them:

Factor out $r$:

So the entire expression simplifies to

which corresponds to answer choice B.