91Ó°ÊÓ

Understanding interval notation is crucial when dealing with inequalities, as it provides a clear and concise way to express ranges of numbers. In the problem provided, we solved for x and found that any number greater than \(\frac{8}{45}\) satisfies the inequality. To represent this in interval notation, we use parentheses to indicate that \(\frac{8}{45}\) is not included in the solution set, and we extend to infinity to show that there's no upper limit to the possible values of x.

The notation \(x \in (\frac{8}{45}, +\infty)\) is read as 'x is any number greater than \(\frac{8}{45}\) but not including \(\frac{8}{45}\) itself.' To improve understanding, remember: use parentheses for numbers that are not included (also called open intervals) and square brackets for numbers that are included (closed intervals) in the solution set.

The distributive property is a foundational concept in algebra that allows us to simplify expressions by multiplying a single term across terms within parentheses. In the exercise, this property is used when we multiply the 6 by each term inside the parentheses (-8x - 1). It works the same way with the 2 and the terms inside (x - 1).

Mathematically, it follows the rule \(a(b + c) = ab + ac\). In our example, it looked like \(6(-8x - 1) = -48x - 6\) and \(2(x - 1) = 2x - 2\). This step is essential for combining like terms afterwards, which helps to simplify the inequality for easier solving. Remember to distribute multipliers over both terms inside the parentheses and to keep the sign of the terms as they are to avoid any errors.

The goal of isolating the variable is to get the variable on one side of the equation or inequality so that the solution becomes clear. In our exercise, we needed to isolate x by performing a series of inverse operations to move other terms to the opposite side of the inequality. We began by adding 43x to both sides in order to move all the x-terms to one side and then added 6 to remove the constant term from the side with the variable.

Finally, we divided by 45 to get the variable alone. This systematic approach of 'undoing' what's been done to the variable, step by step, is the essence of isolating variables. It's a process of reversing operations while maintaining the balance (equality or inequality) of an equation or inequality.