91Ó°ÊÓ

In the realm of algebra, when we talk about rational expressions like \( \frac{2x+1}{(x+2)(1-x)} \), we must always remember that division by zero is undefined. Thus, the denominator of any rational expression cannot be zero. Identifying this fact is crucial in determining the domain, or the set of all possible values of x, for the expression.

Why can't we divide by zero? Well, mathematically, division is the process of finding how many times the divisor can be contained within the dividend. Division by zero implies you're trying to find how many zeroes go into any number, which makes no sense as zero times any number is always zero — it never reaches any other number. Thus, division by zero doesn't produce a meaningful outcome, and so in algebra, we ensure the denominator is never set to zero when finding the domain of rational expressions.

Once we know to avoid zero in the denominator, we look for 'excluded values' which are particular values that x cannot assume because they lead to this forbidden division. In our exercise, we found the excluded values for \( \frac{2x+1}{(x+2)(1-x)} \) by setting the denominator equal to zero and solving for x.

When we did that with \( (x+2)(1-x) = 0 \), we found that x can neither be -2 nor 1, as these would zero out the denominator. These are our excluded values. It’s essential to check each factor of the denominator separately to find all the excluded values, ensuring the rational expression is well-defined for all other numbers. When writing your final answer, it's necessary to communicate that we've not just forgotten about these values; rather, we're purposefully excluding them from our domain.

The final piece of the puzzle in understanding the domain of rational expressions is learning how to express our findings. That's where interval notation comes in; it's a concise way to represent the set of numbers that make up the domain. After excluding the values of x that cannot be in our domain, we represent the remaining values as intervals.

For instance, in our exercise, the domain of expression \( \frac{2x+1}{(x+2)(1-x)} \) excludes -2 and 1. Thus, in interval notation, we write this as \( (-\infty, -2) \cup (-2, 1) \cup (1, \infty) \) which visually shows these gaps around -2 and 1. This notation efficiently shows that our expression works for all real numbers, with the exception of -2 and 1, by using parentheses to indicate that those endpoints are not included in the domain.