Solving rational and real expressions

Question

Assignment:

1^st Rational Expression

 

x² - 36 / 3x

 

Factor Numerator

 

(x + 6)(x - 6) / 3x

 

Evaluate the denominator

 

3x = 0

 

x = 0

 

{x ? R| x ≠ 0}

 

To evaluate the expression, we need to focus on the denominator since it is the only part of the expression that can determine whether it is undefined or not.  To do this, the denominator 3x is evaluated at 0 since it is the value of the denominator that we specifically are looking for.  Solving this would result to x = 0.  Thus, zero would be the excluded value of the domain while the values of the domain can be written as {x ? R| x ≠ 0} read as set of all real numbers excluding zero.

 

The 1^st rational expression's excluded value is 0 since it would result to a zero denominator, which is undefined.

 

2^nd Rational Expression

 

7w - 2 / 16w² - 1

 

Evaluate the denominator

 

16w² - 1 = 0

 

Factor expression at the left hand side of the equation (difference of two squares)

 

(4w + 1)  (4w - 1) = 0

 

4w  + 1 = 0      |          4w - 1 = 0

 

4w = - 1          |           4w = 1

 

4w / 4 = - 1 / 4        |           4w / 4 = 1 / 4

 

w = - ¼           |           w = ¼

 

w = ± ¼

 

{w ? R| x ≠ ± ¼}

 

To evaluate the expression, we again focus on the denominator since it is the only part of the expression should not be equal to 0. Equate the denominator to 0 to find values of w that would result to 0.  Use difference of two squares to factor the expression. Equate the resulting factors to 0.  This would give 4w  + 1 = 0 and 4w - 1 = 0.  Isolate the variable.  Move the 1's to the right side of the equation by either subtracting or adding.  Now that you have 4w = - 1  on the left side and 4w = 1 on the right side.  Now simplify by dividing both sides by 4.  Positive and negative ¼ would be the excluded value of the expression.  The domain would be the set of real numbers excluding positive and negative ¼ read as{w ? R| x ≠ ± ¼}.

 

Domain is the set of real numbers that defines the expression.  Real numbers that makes the expression undefined or resulting to a zero denominator are the excluded values in the domain.  This is common when expressions have denominators, which should not be equal to zero since it is undefined.

 

The 2^nd rational expression's excluded values are ¼ and - ¼.  These are excluded values so it can prevent the denominator from having a value of 0 and making it undefined.

 

Reference:

 

Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY: McGraw-Hill Publishing.

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