Mastery Check Solutions

Show What You Know

The backyard of the Klein house is fenced on three sides. 

The length of the fence is $\frac{f - 5}{f^{2} - 3 f - 10}$ feet. 

The other two sides of the fence are each $\frac{f}{f^{2} + f - 2}$ feet.

Find the total length of the fence.

$fence total = l + 2 w \frac{f - 5}{(f - 5) (f + 2)} + 2 (\frac{f}{(f + 2) (f - 1)}) \frac{1}{(f + 2)} + 2 (\frac{f}{(f + 2) (f - 1)}) LCD : (f + 2) (f - 1) f \neq - 2, 1, 5 \frac{f - 1}{(f + 2) (f - 1)} + \frac{2 f}{(f + 2) (f - 1)} \frac{3 f - 1}{(f + 2) (f - 1)} feet$

Note

Remember to check if it is possible to factor the numerator and simplify further.

Q: What are the restrictions for the denominator?

A: –2, 1, 5

Q: Why do the negative ones not simplify out of the expression?
A: Because only identical binomials can simplify out. (3f –1) and (f – 1) are not identical.

The Klein family hired a surveyor to determine if their yard is large enough for a pool. The ratio of the width to the length of the yard will be used by the surveyor.

Determine the simplified ratio, $\frac{w}{l}$ .

$\frac{\frac{f}{(f + 2) (f - 1)}}{\frac{f - 5}{(f - 5) (f + 2)}} \frac{f}{(f + 2) (f - 1)} \div \frac{f - 5}{(f - 5) (f + 2)}, f \neq - 2, 1, 5 \frac{f}{(f + 2) (f - 1)} \div \frac{f - 5}{(f - 5) (f + 2)} \frac{f}{(f + 2) (f - 1)} \div \frac{1}{(f + 2)} \frac{f}{(f + 2) (f - 1)} \cdot \frac{f + 2}{1} \frac{f}{f - 1}, f \neq - 2, 1, 5$

Note

You can start by writing the ratio as a vertical or horizontal fraction. Remember the variable f in the answer cannot be simplified further because the numerator is a monomial and the denominator is a binomial expression without any common factors.

Q: Why is it not possible to simplify the variable f out of the expression?
A: f and (f – 1) do not have any common factors.

Three members of the Klein family will paint the fence in their backyard.

Joe’s time in minutes	Jack’s time in minutes	Judy’s time in minutes
$\frac{1}{r}$	$\frac{1}{r + 3}$	$\frac{1}{r - 3}$

Find the total time when Joe, Jack, and Judy work together.

$\frac{1}{r} + \frac{1}{r + 3} + \frac{1}{r - 3} LCD : r (r + 3) (r - 3) r \neq \pm 3, 0 \frac{(r + 3) (r - 3)}{r (r + 3) (r - 3)} + \frac{r (r - 3)}{r (r + 3) (r - 3)} + \frac{r (r + 3)}{r (r + 3) (r - 3)} Numerator : r^{2} - 9 + r^{2} - 3 r + r^{2} + 3 r 3 r^{2} - 9 \frac{3 (r^{2} - 3)}{r (r + 3) (r - 3)}, r \neq \pm 3, 0$

Note

In a later lesson you will learn how to solve for the value of r.

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Determine the least common denominator (LCD) of a rational expression.
Add and subtract rational expressions.
Simplify complex fractions using addition, subtraction, multiplication, and division.
Apply rational expressions to word problems such as geometric probabilities and efficiency ratios.