The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

Teaching Notes 6.4: Identifying Restrictions on the Variable

The value of the denominator of any fraction cannot equal 0. When students work with algebraic

fractions, they must identify those values that would make the denominator equal to 0 and

exclude them from the solution. This concept, placing a restriction on a variable, can be

confusing, especially because students must factor the denominator and identify the values that

would make the denominator equal to 0.

1. Explain to your students that a fraction such as

5

0

is undefined. To show your students why

this is so, provide examples of division problems such as

− 12

3

=−4. Note that students can

check this answer by multiplying− 4 · 3 =−12. Now provide an example such as

5

0

and ask

students to provide an answer that they think is correct. One common response is 5. Note

that 0· 5 =5. The answer does not check, nor will any other. Using this reasoning, students

will realize that division by 0 is impossible.

2. Review the information and examples on the worksheet with your students. Emphasize that
   if there is a variable in the denominator, students must restrict the variables in the denom-
   inator so that the denominator does not equal zero. Discuss the procedure for identifying
   restrictions on variables in detail. Note that if the highest power of the variable is 1, students
   should write and solve an inequality that states that the denominator does not equal 0. If the
   highest power of the variable is 2 or more, students must factor the denominator. If at least
   one of the factors is equal to 0, the denominator is equal to 0. Discuss the examples, making
   certain that your students understand how to solve the inequalities.

EXTRA HELP:

The numerator has no effect on the restrictions of the variable.

ANSWER KEY:

(1)x= 4 (2)x= 0 (3)x=1;x=− 1 (4)x=−4;x= 1

(5)x=−1;x= 11 (6)x=−7;x= 1 (7)x=0;x= 1 (8)x=−8;x=− 5

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(Challenge)Thefractionequals1,providedthatx=2. Whenx=2, the denominator is equal

to 0 and the fraction is undefined.

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230 THE ALGEBRA TEACHER’S GUIDE