Lesson 5 Just Act Rational Solidify Understanding

Learning Focus

Add, subtract, multiply, and divide rational expressions.

How are the operations performed on rational expressions like operations performed on rational numbers?

What ideas about fractions are revealed when we change forms?

Open Up the Math: Launch, Explore, Discuss

In the previous lesson you saw how connecting rational numbers can help us to think about rational expressions. In this task, we’ll extend that work to consider operations on rational expressions.

1.

Let’s begin with multiplication. In the table there are missing descriptions and missing parts of expressions. Use the process for multiplying rational numbers to complete the descriptions of the process and perform the operations to find equivalent rational expressions.

Description of the Procedure:	Example Using Numbers	Rational Expression A	Rational Expression B
Given:	$\frac{3}{4} \cdot \frac{5}{6}$	$\frac{x (x - 3)}{(x + 1)} \cdot \frac{5}{x^{2}}$	$\frac{(x + 1) (x - 2)}{(x + 2)} \cdot \frac{(x + 5)}{(x - 2) (x + 2)}$
	$\frac{3 \cdot 5}{4 \cdot 6}$
	$\frac{3 \cdot 5}{2 \cdot 2 \cdot 2 \cdot 3}$
Write an equivalent expression:	$\frac{5}{8}$	$\frac{5 (x - 3)}{x (x + 1)}$ or $\frac{5 x - 15}{x (x + 1)}$	$\frac{(x + 1) (x + 5)}{{(x + 2)}^{2}}$ or $\frac{x^{2} + 6 x + 5}{{(x + 2)}^{2}}$

2.

In multiplication, does it matter if the common factors are divided out before or after multiplying? Why?

Now let’s try the same process with division.

3.

Complete the table by filling in the missing descriptions or steps for dividing the rational expressions.

Description of the Procedure:	Example Using Numbers	Rational Expression A	Rational Expression B
Given:		$\frac{(x - 2)}{(x + 2)} \div \frac{(x + 5)}{x (x + 2)}$	$\frac{(x + 1) (x - 6)}{(x + 2)} \div \frac{(x + 1)}{(x - 3) (x + 2)}$
	$\frac{3}{4} \cdot \frac{6}{5}$
Multiply and divide out any common factors:	$\frac{3 \cdot 6}{4 \cdot 5} = \frac{3 \cdot 2 \cdot 3}{2 \cdot 2 \cdot 5}$
	$\frac{9}{10}$	$\frac{x (x - 2)}{(x + 5)}$ or $\frac{x^{2} - 2 x}{(x + 5)}$	$(x - 3) (x - 6)$

4.

How could you check your answer after performing an operation on a pair of rational expressions?

Are you ready for addition? Try it!

5.

Complete the table by filling in the missing descriptions or steps for adding the rational expressions.

Description of the Procedure:	Example Using Numbers	Rational Expression A
Given:	$\frac{2}{3} + \frac{1}{7}$	$\frac{3}{(x + 7)} + \frac{4}{(x - 4)}$
Determine the factors needed for a common denominator	$\frac{2}{3} (\frac{7}{7}) + \frac{1}{7} (\frac{3}{3})$	$\frac{3}{(x + 7)} (\frac{x - 4}{x - 4}) + \frac{4}{(x - 4)} (\frac{x + 7}{x + 7})$
	$\frac{14}{21} + \frac{3}{21}$
	$\frac{14 + 3}{21}$
Divide out any common factors	$\frac{17}{21}$	$\frac{7 x + 16}{(x + 7) (x - 4)}$

6.

After writing both terms with a common denominator, are they equivalent to the original terms? Explain.

7.

Complete the table by filling in the missing descriptions and steps for adding the rational expressions.

Description of the Procedure:	Rational Expression B
Given:	$\frac{2 x - 3}{(x + 3)} + \frac{x + 5}{(x - 2)}$
Determine the factors needed for a common denominator


Divide out any common factors and write the equivalent expression	$\frac{3 x^{2} + x + 21}{(x - 2) (x + 3)}$

8.

Is it possible to get an answer to an addition problem that contains common factors that should be divided out? If so, how can you tell?

9.

At long last, we have subtraction. Complete the table by filling in the missing descriptions and steps for subtracting the rational expressions.

Description of the Procedure:	Example Using Numbers	Rational Expression A
Given:	$\frac{7}{8} - \frac{3}{5}$	$\frac{3 x + 1}{(x + 5)} - \frac{x - 4}{(x - 2)}$
	$\frac{7}{8} (\frac{5}{5}) - \frac{3}{5} (\frac{8}{8})$
	$\frac{35}{40} - \frac{24}{40}$
	$\frac{35 - 24}{40}$
Divide out any common factors and write the equivalent expression	$\frac{11}{40}$	$\frac{2 x^{2} - 6 x + 18}{(x + 5) (x - 2)}$ or $\frac{2 (x^{2} - 3 x + 9)}{(x + 5) (x - 2)}$

10.

What strategies will you use to be sure that you don’t make sign errors when subtracting?

Ready for More?

Is $\frac{1}{a + b} = \frac{1}{a} + \frac{1}{b}$ ? Explain.

Takeaways

Operations on Rational Expressions

Steps for Multiplication	Steps for Division	Steps for Addition	Steps for Subtraction

Lesson Summary

In this lesson, we learned that performing operations on rational expressions is just like performing operations on rational numbers. Multiplication is performed by multiplying the numerators together, multiplying the denominators together, and dividing out any common factors. Division is performed by inverting the divisor and then multiplying the two fractions. Addition and subtraction require obtaining a common denominator and then combining the numerators into one fraction with the common denominator.

Retrieval

1.

Use the triangle to find the values.

a.

$\sin A = \underset{―}{}$

b.

$\cos A = \underset{―}{}$

c.

$\tan A = \underset{―}{}$

d.

$\sin B = \underset{―}{}$

e.

$\cos B = \underset{―}{}$

f.

$\tan B = \underset{―}{}$

2.

Identify the value(s) for which the expression is undefined: $\frac{x + 8}{x - 2}$