91Ó°ÊÓ

One efficient way to express the solution set of inequalities is through interval notation. This method uses parentheses or brackets to indicate which numbers are included in a set. For instance, in the expression \((-2, 0]\), the parentheses around -2 show that -2 is not part of the set. However, the square bracket around 0 indicates that 0 is included.

This is essential because sometimes an inequality might equal zero, and other times it might not. We use interval notation to clearly express these inclusions and exclusions. In mathematical solutions, interval notation is crucial, as it gives a precise understanding of the range of values that satisfy an inequality.

Interval notation stands as a concise way to convey ranges without needing lengthy explanations.

Critical points are key in solving rational inequalities. They help us understand the behavior of a function at specific places. A critical point occurs in two cases: when the numerator of the expression equals zero, or when the denominator equals zero, which causes the expression to become undefined.

For example, when we look at the inequality \(\frac{-2x}{x + 2} \leq 0\), we find critical points by setting the numerator and denominator separately equal to zero.

• Numerator: \(-2x = 0\) gives the critical point \(x = 0\).
• Denominator: \(x + 2 = 0\) gives the critical point \(x = -2\).

We then use these critical points to segment the number line and investigate which sections meet the original inequality. They are pivotal in determining where the solution begins and ends.

The real number line is a visual representation that helps in understanding how numbers are arranged. By plotting critical points on this line, we can segment it into different regions for evaluation.

In our exercise, the real number line is split by the critical points \(x = -2\) and \(x = 0\), forming three segments: \(x < -2\), \(-2 < x < 0\), and \(0 < x\).

When we test these regions against our inequality, we observe which sections satisfy the condition. The real number line graphically aids in visualizing which sections include valid solutions and which do not. Using this approach makes it much easier to see the relationship between the values and the mathematical inequality.

The solution set refers to the set of values that satisfy a given inequality or equation. For this exercise, after testing each region using the real number line, the values that meet the inequality \(\frac{-2x}{x+2} \leq 0\) were found.

In this case, the interval \((-2, 0]\) emerged as the solution set. To explain further:

• The interval starts just after -2, indicated by the parenthesis, which excludes -2 since it makes the denominator \(x + 2\) zero, thus undefined.
• The interval ends at 0, included as denoted by the bracket, because it satisfies the inequality.

Understanding the solution set is crucial. It tells us exactly which values our function can take, capturing all possible solutions for the inequality at hand.