Algebra 2 Unit 2: Rational Expressions and Equations

Simplifying and combining rational expressions, solving rational equations, extraneous solutions, asymptotes and holes. 12 questions. Have a go at each one on paper before you tap "Show me how".

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Question 1

Simplifying rational expressionsNo calculator

Simplify $\frac{x^2-9}{x^2+5x+6}$.

1. $\frac{x+3}{x+2}$
2. $\frac{x-3}{x+3}$
3. $\frac{-9}{5x+6}$
4. $\frac{x-3}{x+2}$

Question 2

Multiplying rational expressionsNo calculator

Simplify $\frac{x}{x+1}\cdot\frac{x^2-1}{x^2}$.

1. $\frac{x-1}{x^{2}}$
2. $x-1$
3. $\frac{x+1}{x}$
4. $\frac{x-1}{x}$

Question 3

Adding rational expressionsNo calculator

Combine $\frac{2}{x-3}+\frac{5}{x+2}$ into one fraction over a common denominator.

1. $\frac{7x-11}{(x-3)(x+2)}$
2. $\frac{7x-11}{x^{2}-6}$
3. $\frac{7}{2x-1}$
4. $\frac{7x+19}{(x-3)(x+2)}$

Question 4

Rational equationsNo calculator

Solve $\frac{3}{x}+\frac{1}{2}=\frac{2}{x}$.

1. $x=2$
2. $x=-\frac{1}{2}$
3. $x=\frac{1}{2}$
4. $x=-2$

Question 5

Extraneous solutionsNo calculator

Solve $\frac{x}{x-2}=\frac{2}{x-2}+3$.

1. $x=2$
2. $x=4$
3. $x=-2$
4. No solution

Question 6

AsymptotesNo calculator

What are the vertical asymptotes of $f(x)=\frac{x+1}{x^2-4x}$?

1. $x=0 \text{ and } x=4$
2. $x=-1$
3. $x=0 \text{ and } x=-4$
4. $x=4$

Question 7

HolesNo calculator

The graph of $f(x)=\frac{x^2-x-6}{x-3}$ has a hole. Where is it?

1. $(3,5)$
2. $(3,-2)$
3. $(3,0)$
4. $(-2,0)$

Question 8

AsymptotesNo calculator

What is the horizontal asymptote of $f(x)=\frac{4x^2-1}{2x^2+x}$?

1. $y=4$
2. $y=0$
3. $y=2$
4. $y=\frac{1}{2}$

Question 9

Rational modelsNo calculator

One pipe fills a tank in 6 hours and a second pipe fills it in 3 hours. How many hours do they take working together?

1. $4.5$
2. $3$
3. $9$
4. $2$

Question 10

Inverse variationNo calculator

$y$ varies inversely with $x$, and $y=8$ when $x=3$. What is $y$ when $x=12$?

1. $4.5$
2. $32$
3. $6$
4. $2$

Question 11

Complex fractionsNo calculator

Simplify $\dfrac{1+\frac{1}{x}}{1-\frac{1}{x^2}}$.

1. $\frac{x}{x-1}$
2. $\frac{x+1}{x-1}$
3. $\frac{1}{x-1}$
4. $x$

Question 12

Rational equationsNo calculator

Solve $\frac{1}{x}+\frac{1}{x+2}=\frac{4}{x(x+2)}$.

1. $x=1$
2. $x=0$
3. $x=2$
4. $x=-1$