A non-permissible value (sometimes called a restriction) is a value of the variable that would make the denominator of a fraction equal to zero. Since division by zero is undefined, we must exclude these values from the domain of the expression.

When finding non-permissible values, you don’t need to simplify the numerator; only the denominator matters. The steps are:

1. Identify the denominator.
2. Set it equal to zero.
3. Solve for x and exclude those values.

Practice Problems

1. Find all non-permissible values of 2/x
2. Find all non-permissible values of (x + 2) / (x − 3)
3. Find all non-permissible values of (x² − 9) / (x² − 7x + 12)
4. Find all non-permissible values of 5 / [(x − 1)(x + 4)]
5. Find all non-permissible values of (x + 5) / (x² + 6x + 9)

Solutions

1. The denominator is x.
   Division by zero is undefined, so we set x = 0 and exclude it. x ≠ 0

2. The denominator is x − 3.
   Setting x − 3 = 0 gives x = 3. x ≠ 3

3. The denominator is x² − 7x + 12. Factor it: x² − 7x + 12 = (x − 4)(x − 3) Each factor could be zero, so exclude x = 4 and x = 3.  x ≠ 4, 3

4. The denominator is (x − 1)(x + 4).
   Set each factor equal to zero: x − 1 = 0 ⇒ x = 1. x + 4 = 0 ⇒ x = −4.  x ≠ 1, −4

5. The denominator is x² + 6x + 9. Factor it: x² + 6x + 9 = (x + 3)(x + 3) = (x + 3)² Only x = −3 makes the denominator zero.  x ≠ −3