Simplify the following and determine if the simplified expression is equivalent to the original:
Answer
We factor out the common factor of x and cancel to get the simplified form:
The simplified expression is not equivalent to the original, since the simplified expression can be evaluated at x = 0 but the original expression can't be evaluated at x = 0. Nice try, expression, but you need to wake up early in the morning to pull one over on us. We do like to sleep in until at least 10:30 on Sundays, though.
Example 2
Simplify the following and determine if the simplified expression is equivalent to the original:
Answer
Factor out (x + 2) from the numerator and denominator to find the simplified form:
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The simplified expression and original expression are equivalent. Neither can be evaluated at x = -2, and the two expressions agree on all other values of x. They have differing political views, but they try not to talk about it at the dinner table.
Example 3
Simplify the following and determine if the simplified expression is equivalent to the original:
Answer
Factor as
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The original expression can't be evaluated at , but the simplified expression can be evaluated at . This means the two expressions are not equivalent. Bummer. We had such high hopes for these two.
Example 4
Simplify the following and determine if the simplified expression is equivalent to the original:
Answer
Pull out the common factor xy from the numerator and denominator to get x + y.
This expression can be evaluated for x = 0 and y = 0. The original expression can't be evaluated when x or y is 0, so the two expressions aren't equivalent.
Example 5
Simplify the following and determine if the simplified expression is equivalent to the original:
Answer
The expression factors as , which simplifies to x + 1. While x + 1 can be evaluated for any value of x, the original expression can't be evaluated at x = 0 or , so the two expressions aren't equivalent.
.
.
, but the simplified expression can be evaluated at 
. This means the two expressions are not equivalent. Bummer. We had such high hopes for these two.
, which simplifies to x + 1. While x + 1 can be evaluated for any value of x, the original expression can't be evaluated at x = 0 or 
, so the two expressions aren't equivalent.